complexGEcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💅 complexGEcomputational(3) — LAPACK 💅

bestie, this is the complexGEcomputational page from LAPACK, and it's giving ✨ complex GE matrix computational functions ✨. We're in part 1, so buckle up. 🫶

✨ NAME ✨

complexGEcomputational — complex 💖

🌸 SYNOPSIS 🌸

Functions

subroutine cgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO) — CGEBAK
subroutine cgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO) — CGEBAL
subroutine cgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) — CGEBD2 is the unblocked girlie that reduces a general matrix to bidiagonal form. ✨
subroutine cgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) — CGEBRD
subroutine cgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO) — CGECON
subroutine cgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) — CGEEQU
subroutine cgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) — CGEEQUB
subroutine cgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO) — CGEHD2 takes a general square matrix to upper Hessenberg form, unblocked algorithm energy. 💅
subroutine cgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO) — CGEHRD
subroutine cgelq2 (M, N, A, LDA, TAU, WORK, INFO) — CGELQ2 computes the LQ factorization of a general rectangular matrix, unblocked algorithm style. 🦋
subroutine cgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — CGELQF
subroutine cgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO) — CGEMQRT
subroutine cgeql2 (M, N, A, LDA, TAU, WORK, INFO) — CGEQL2 computes the QL factorization of a general rectangular matrix, unblocked algorithm vibes. 🌸
subroutine cgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — CGEQLF
subroutine cgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, RWORK, INFO) — CGEQP3
subroutine cgeqr2 (M, N, A, LDA, TAU, WORK, INFO) — CGEQR2 computes the QR factorization of a general rectangular matrix, unblocked algorithm. ✨
subroutine cgeqr2p (M, N, A, LDA, TAU, WORK, INFO) — CGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements, unblocked algorithm. 💖
subroutine cgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — CGEQRF
subroutine cgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO) — CGEQRFP
subroutine cgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO) — CGEQRT
subroutine cgeqrt2 (M, N, A, LDA, T, LDT, INFO) — CGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. 👑
recursive subroutine cgeqrt3 (M, N, A, LDA, T, LDT, INFO) — CGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. 🔥
subroutine cgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) — CGERFS
subroutine cgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) — CGERFSX
subroutine cgerq2 (M, N, A, LDA, TAU, WORK, INFO) — CGERQ2 computes the RQ factorization of a general rectangular matrix, unblocked algorithm. 🎀
subroutine cgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — CGERQF
subroutine cgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO) — CGESVJ
subroutine cgetf2 (M, N, A, LDA, IPIV, INFO) — CGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm). 💀
subroutine cgetrf (M, N, A, LDA, IPIV, INFO) — CGETRF
recursive subroutine cgetrf2 (M, N, A, LDA, IPIV, INFO) — CGETRF2
subroutine cgetri (N, A, LDA, IPIV, WORK, LWORK, INFO) — CGETRI
subroutine cgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — CGETRS
subroutine chgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) — CHGEQZ
subroutine cla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — CLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds. 🌈
real function cla_gercond_c (TRANS, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK) — CLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices. 🧚
real function cla_gercond_x (TRANS, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK) — CLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices. 💜
subroutine cla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) — CLA_GERFSX_EXTENDED
real function cla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF) — CLA_GERPVGRW multiplies a square real matrix by a complex matrix. 🍓
recursive subroutine claqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, REC, INFO) — CLAQZ0
subroutine claqz1 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ, NZ, ZSTART, Z, LDZ) — CLAQZ1
recursive subroutine claqz2 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHA, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, RWORK, REC, INFO) — CLAQZ2
subroutine claqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, ALPHA, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO) — CLAQZ3
subroutine claunhr_col_getrfnp (M, N, A, LDA, D, INFO) — CLAUNHR_COL_GETRFNP
recursive subroutine claunhr_col_getrfnp2 (M, N, A, LDA, D, INFO) — CLAUNHR_COL_GETRFNP2
subroutine ctgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO) — CTGEVC
subroutine ctgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO) — CTGEXC

💖 Detailed Description 💖

okay so this is literally the group chat of complex computational functions for GE matrices. 💅

🫶 Function Documentation 🫶

💅 subroutine cgebak

subroutine cgebak (character JOB, character SIDE, integer N, integer ILO,
    integer IHI, real, dimension( * ) SCALE, integer M, complex, dimension(
    ldv, * ) V, integer LDV, integer INFO)

CGEBAK

✨ Purpose ✨

CGEBAK forms the right or left eigenvectors of a complex general matrix by doing a backward transformation on the computed eigenvectors of the balanced matrix that CGEBAL output. Basically it un-balances your eigenvectors so they match the original matrix. 👑

🌸 Parameters 🌸

JOB — JOB is CHARACTER*1. Specifies the type of backward transformation required:
- 'N': do nothing, return immediately;
- 'P': do backward transformation for permutation only;
- 'S': do backward transformation for scaling only;
- 'B': do backward transformations for both permutation and scaling. JOB must be the same as the argument JOB supplied to CGEBAL.
SIDE — SIDE is CHARACTER*1.
- 'R': V contains right eigenvectors;
- 'L': V contains left eigenvectors.
N — N is INTEGER. The number of rows of the matrix V. N >= 0.
ILO — ILO is INTEGER
IHI — IHI is INTEGER. The integers ILO and IHI determined by CGEBAL. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
SCALE — SCALE is REAL array, dimension (N). Details of the permutation and scaling factors, as returned by CGEBAL.
M — M is INTEGER. The number of columns of the matrix V. M >= 0.
V — V is COMPLEX array, dimension (LDV,M). On entry, the matrix of right or left eigenvectors to be transformed, as returned by CHSEIN or CTREVC. On exit, V is overwritten by the transformed eigenvectors.
LDV — LDV is INTEGER. The leading dimension of the array V. LDV >= max(1,N).
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💅 subroutine cgebal

subroutine cgebal (character JOB, integer N, complex, dimension( lda, * ) A,
    integer LDA, integer ILO, integer IHI, real, dimension( * ) SCALE,
    integer INFO)

CGEBAL

✨ Purpose ✨

CGEBAL balances a general complex matrix A. First, it permutes A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; second, it applies a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional.

Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors. ✨

🌸 Parameters 🌸

JOB — JOB is CHARACTER*1. Specifies the operations to be performed on A:
- 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 for i = 1,...,N;
- 'P': permute only;
- 'S': scale only;
- 'B': both permute and scale.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced. See Further Details.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
ILO — ILO is INTEGER
IHI — IHI is INTEGER. ILO and IHI are set to integers such that on exit A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
SCALE

SCALE is a REAL array, dimension (N) and it's literally holding all the receipts 📋 — the deets of the permutations and scaling factors that got applied to A. So if P(j) is the index of the row and column that got swapped with row and column j, and D(j) is the scaling factor slapped onto row and column j, then bestie, it's giving:

SCALE(j) = P(j)    for j = 1,...,ILO-1
         = D(j)    for j = ILO,...,IHI
         = P(j)    for j = IHI+1,...,N.

And the order the interchanges go down? It's N to IHI+1, then 1 to ILO-1. Not the other way around, okurrr 💅

INFO — this one's an INTEGER and it's the vibe check on whether it worked:
- = 0: successful exit, we ate and left no crumbs ✨
- < 0: if INFO = -i, then the i-th argument had an illegal value. Yeah, you passed something cursed and it's calling you out 💀

🎀 Author 🎀

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

✨ Further Details ✨

So the permutations are made up of row and column interchanges that put the matrix into this exact form:

         ( T1   X   Y  )
P A P =  (  0   B   Z  )
         (  0   0   T2 )

Where T1 and T2 are upper triangular matrices whose eigenvalues sit right along the diagonal — no cap. The column indices ILO and IHI mark the starting and ending columns of the submatrix B. And balancing? It's the glow-up of applying a diagonal similarity transformation inv(D) * B * D to make the 1-norms of each row of B and its corresponding column nearly equal. Main character energy for both of them, fr fr. The output matrix comes out looking like:

    ( T1     X*D          Y    )
    (  0  inv(D)*B*D  inv(D)*Z ).
    (  0      0           T2   )

All the info about the permutations P and the diagonal matrix D gets returned in the vector SCALE. Slay.

This subroutine is based on the EISPACK routine CBAL.

Modified by Tzu-Yi Chen, Computer Science Division, University of California at Berkeley, USA 💜

🦋 subroutine cgebd2 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( * ) WORK, integer INFO)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. (Unblocked = the no-frills, straight-through version, iykyk 🧚)

Purpose ✨

CGEBD2 reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation:

Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. Simple as that 💅

Parameters 🎀

M — INTEGER. The number of rows in the matrix A. M >= 0.
N — INTEGER. The number of columns in the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- On entry: the m by n general matrix to be reduced.
- On exit:
  - if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors;
  - if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
- See Further Details, bestie 👀
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
D — REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E — REAL array, dimension (min(M,N)-1). The off-diagonal elements of the bidiagonal matrix B:
- if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
- if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
TAUP — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
WORK — COMPLEX array, dimension (max(M,N)).
INFO — INTEGER:
- = 0: successful exit 🌸
- < 0: if INFO = -i, the i-th argument had an illegal value. (Not you doing that 🙈)

Author 💜

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details 💫

The matrices Q and P are represented as products of elementary reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

Where tauq and taup are complex scalars, and v and u are complex vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

Where tauq and taup are complex scalars, v and u are complex vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

  (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  (  v1  v2  v3  v4  v5 )

Where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). Obsessed with how tidy that is tbh 🦋

🌈 subroutine cgebrd (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGEBRD

Purpose ✨

CGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation:

Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. Same energy as its little sibling, just blocked and glowier 💅

Parameters 🎀

M — INTEGER. The number of rows in the matrix A. M >= 0.
N — INTEGER. The number of columns in the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- On entry: the M-by-N general matrix to be reduced.
- On exit:
  - if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors;
  - if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors.
- See Further Details, iykyk 👀
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
D — REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E — REAL array, dimension (min(M,N)-1). The off-diagonal elements of the bidiagonal matrix B:
- if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
- if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
TAUP — COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. (It's telling you how much room it actually wanted 🫶)
LWORK — INTEGER. The length of the array WORK. LWORK >= max(1,M,N). For optimum performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So don't be shocked when it stays silent, bestie 💀

INFO — INTEGER:
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value.

Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details:

okay bestie, so the matrices Q and P are like products of elementary reflectors, and it's giving very demure, very mindful linear algebra. 💅

If m >= n, then

Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, then

Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by these examples, no cap:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

  (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  (  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

✨ CGECON ✨

   subroutine cgecon (character NORM, integer N, complex, dimension( lda, * )
       A, integer LDA, real ANORM, real RCOND, complex, dimension( * ) WORK,
       real, dimension( * ) RWORK, integer INFO)

Purpose:

CGECON estimates the reciprocal of the condition number of a general complex matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by CGETRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) )

Parameters

NORM — NORM is CHARACTER*1. Specifies whether the 1-norm condition number or the infinity-norm condition number is required:
- = '1' or 'O': 1-norm;
- = 'I': Infinity-norm.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
ANORM — ANORM is REAL. If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK — WORK is COMPLEX array, dimension (2*N).
RWORK — RWORK is REAL array, dimension (2*N).
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌸 CGEEQU 🌸

   subroutine cgeequ (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, real, dimension( * ) R, real, dimension( * ) C, real
       ROWCND, real COLCND, real AMAX, integer INFO)

Purpose:

CGEEQU computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

R(i) and C(j) are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The M-by-N matrix whose equilibration factors are to be computed.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
R — R is REAL array, dimension (M). If INFO = 0 or INFO > M, R contains the row scale factors for A.
C — C is REAL array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — ROWCND is REAL. If INFO = 0 or INFO > M, ROWCND contains the ratio of the smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R.
COLCND — COLCND is REAL. If INFO = 0, COLCND contains the ratio of the smallest C(i) to the largest C(i). If COLCND >= 0.1, it is not worth scaling by C.
AMAX — AMAX is REAL. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, and i is
  - <= M: the i-th row of A is exactly zero
  - > M: the (i-M)-th column of A is exactly zero

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🦋 CGEEQUB 🦋

   subroutine cgeequb (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, real, dimension( * ) R, real, dimension( * ) C, real
       ROWCND, real COLCND, real AMAX, integer INFO)

Purpose:

CGEEQUB computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most the radix.

R(i) and C(j) are restricted to be a power of the radix between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice.

This routine differs from CGEEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The M-by-N matrix whose equilibration factors are to be computed.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
R — R is REAL array, dimension (M). If INFO = 0 or INFO > M, R contains the row scale factors for A.
C — C is REAL array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — ROWCND is REAL. If INFO = 0 or INFO > M, ROWCND contains the ratio of the smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R.
COLCND — COLCND is REAL. If INFO = 0, COLCND contains the ratio of the smallest C(i) to the largest C(i). If COLCND >= 0.1, it is not worth scaling by C.
AMAX — AMAX is REAL. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, and i is
  - <= M: the i-th row of A is exactly zero
  - > M: the (i-M)-th column of A is exactly zero

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 CGEHD2 💜

   subroutine cgehd2 (integer N, integer ILO, integer IHI, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex,
       dimension( * ) WORK, integer INFO)

CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

CGEHD2 reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q**H * A * Q = H .

Parameters

N — N is INTEGER. The order of the matrix A. N >= 0.
ILO — ILO is INTEGER.
IHI — IHI is INTEGER. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= max(1,N).
A

`A` 🦋

A is a COMPLEX array, dimension (LDA,N).

On entry, it's the n by n general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal — plus the array TAU — represent the unitary matrix Q as a product of elementary reflectors. See Further Details for the whole tea. 💅

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). No cap, fr. ✨

`TAU`

TAU is a COMPLEX array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).

`WORK`

WORK is a COMPLEX array, dimension (N).

`INFO`

INFO is INTEGER.

= 0: successful exit ✨ (we ate, we left no crumbs)
< 0: if INFO = -i, the i-th argument had an illegal value. 💀

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

✨ Further Details ✨

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). iykyk 🧚

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). literally a glow-up in matrix form 🌈

💖 `subroutine cgehrd` 💖

subroutine cgehrd (integer N, integer ILO, integer IHI, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex,
    dimension( * ) WORK, integer LWORK, integer INFO)

CGEHRD 👑

🎯 Purpose:

CGEHRD reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q**H * A * Q = H. That's it, that's the whole flex. 🔥

Parameters 🌸

`N`

N is INTEGER. The order of the matrix A. N >= 0.

`ILO`

ILO is INTEGER.

`IHI`

IHI is INTEGER.

It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. And the rules are: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 💅

`A`

A is a COMPLEX array, dimension (LDA,N).

On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).

`TAU`

TAU is a COMPLEX array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. 🫶

`WORK`

WORK is a COMPLEX array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

`LWORK`

LWORK is INTEGER. The length of the array WORK. LWORK >= max(1,N).

For good performance, LWORK should generally be larger. (she's an overachiever, let her cook 🍓)

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

`INFO`

INFO is INTEGER.

= 0: successful exit ✨
< 0: if INFO = -i, the i-th argument had an illegal value. 😭

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔍 Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

This file is a slight modification of LAPACK-3.0's CGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See CLAHR2.) 📚

💜 `subroutine cgelq2` 💜

subroutine cgelq2 (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer INFO)

CGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm. 🎀

🎯 Purpose:

CGELQ2 computes an LQ factorization of a complex m-by-n matrix A:

A = ( L 0 ) *  Q

where:

Q is a n-by-n orthogonal matrix;
L is a lower-triangular m-by-m matrix;
0 is a m-by-(n-m) zero matrix, if m < n.

Parameters 🌸

`M`

M is INTEGER. The number of rows of the matrix A. M >= 0.

`N`

N is INTEGER. The number of columns of the matrix A. N >= 0.

`A`

A is a COMPLEX array, dimension (LDA,N).

On entry, the m by n matrix A. On exit, the elements on and below the diagonal of the array contain the m by min(m,n) lower trapezoidal matrix L (L is lower triangular if m <= n); the elements above the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

`TAU`

TAU is a COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).

`WORK`

WORK is a COMPLEX array, dimension (M).

`INFO`

INFO is INTEGER.

= 0: successful exit ✨
< 0: if INFO = -i, the i-th argument had an illegal value 💀

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔍 Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(i,i+1:n), and tau in TAU(i). 🦋

🎀 `subroutine cgelqf` 🎀

subroutine cgelqf (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer LWORK, integer INFO)

CGELQF 👑

🎯 Purpose:

CGELQF computes an LQ factorization of a complex M-by-N matrix A:

A = ( L 0 ) *  Q

where:

Q is a N-by-N orthogonal matrix;
L is a lower-triangular M-by-M matrix;
0 is a M-by-(N-M) zero matrix, if M < N.

Parameters 🌸

`M`

M is INTEGER. The number of rows of the matrix A. M >= 0.

`N`

N is INTEGER. The number of columns of the matrix A. N >= 0.

`A`

A is a COMPLEX array, dimension (LDA,N).

On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the m-by-min(m,n) lower trapezoidal matrix L (L is lower triangular if m <= n); the elements above the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

`TAU`

TAU is a COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).

`WORK`

WORK is a COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🫶

`LWORK`

LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,M).

For optimum performance LWORK >= M*NB, where NB is the optimal blocksize.

...and yeah, if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. It's giving "just tell me how much room you need, no drama" 💅

INFO — INFO is INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 Further Details

The matrix Q is represented as a product of elementary reflectors

   Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in A(i,i+1:n), and tau in TAU(i).

🦋 subroutine cgemqrt

subroutine cgemqrt (character SIDE, character TRANS, integer M, integer N,
    integer K, integer NB, complex, dimension( ldv, * ) V, integer LDV,
    complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, *
    ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CGEMQRT — ok so this one's the glow-up version, fr fr ✨

💖 Purpose

CGEMQRT overwrites the general complex M-by-N matrix C with

                SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q C            C Q
TRANS = 'C':    Q**H C            C Q**H

where Q is a complex orthogonal matrix defined as the product of K elementary reflectors:

      Q = H(1) H(2) . . . H(K) = I - V T V**H

generated using the compact WY representation as returned by CGEQRT. 🫶

Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.

🎀 Parameters

SIDE — SIDE is CHARACTER*1
- = 'L': apply Q or Q**H from the Left;
- = 'R': apply Q or Q**H from the Right.
TRANS — TRANS is CHARACTER*1
- = 'N': No transpose, apply Q;
- = 'C': Conjugate transpose, apply Q**H.
M — M is INTEGER. The number of rows of the matrix C. M >= 0.
N — N is INTEGER. The number of columns of the matrix C. N >= 0.
K — K is INTEGER. The number of elementary reflectors whose product defines the matrix Q.
- If SIDE = 'L', M >= K >= 0;
- if SIDE = 'R', N >= K >= 0.
NB — NB is INTEGER. The block size used for the storage of T. K >= NB >= 1. This must be the same value of NB used to generate T in CGEQRT. 👀 like, don't switch it up, bestie, it has to match.
V — V is COMPLEX array, dimension (LDV,K). The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CGEQRT in the first K columns of its array argument A.
LDV — LDV is INTEGER. The leading dimension of the array V.
- If SIDE = 'L', LDA >= max(1,M);
- if SIDE = 'R', LDA >= max(1,N).
T — T is COMPLEX array, dimension (LDT,K). The upper triangular factors of the block reflectors as returned by CGEQRT, stored as a NB-by-N matrix.
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
C — C is COMPLEX array, dimension (LDC,N).
- On entry, the M-by-N matrix C.
- On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M).
WORK — WORK is COMPLEX array. The dimension of WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
INFO — INFO is INTEGER
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈

🌈 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 subroutine cgeql2

subroutine cgeql2 (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer INFO)

CGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm. 💅 the lil' unblocked one, doing its best, and honestly? slaying.

⭐ Purpose

CGEQL2 computes a QL factorization of a complex m by n matrix A:

A = Q * L.

🌸 Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N).
- On entry, the m by n matrix A.
- On exit, if m >= n, the lower triangle of the subarray A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
- if m <= n, the elements on and below the (n-m)-th superdiagonal contain the m by n lower trapezoidal matrix L;
- the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is COMPLEX array, dimension (N).
INFO — INFO is INTEGER
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💜 Further Details

The matrix Q is represented as a product of elementary reflectors

   Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(1:m-k+i-1,n-k+i), and tau in TAU(i).

🎀 subroutine cgeqlf

subroutine cgeqlf (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer LWORK, integer INFO)

CGEQLF — same energy as CGEQL2 but it got the blocked glow-up, no cap 🔥

✨ Purpose

CGEQLF computes a QL factorization of a complex M-by-N matrix A:

A = Q * L.

🦋 Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N).
- On entry, the M-by-N matrix A.
- On exit,
- if m >= n, the lower triangle of the subarray A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
- if m <= n, the elements on and below the (n-m)-th superdiagonal contain the M-by-N lower trapezoidal matrix L;
- the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👀 she tells you the tea on exit.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO — INFO is INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 💀

🌈 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💫 Further Details

The matrix Q is represented as a product of elementary reflectors

   Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(1:m-k+i-1,n-k+i), and tau in TAU(i).

🍓 subroutine cgeqp3

subroutine cgeqp3 (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU,
    complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
    integer INFO)

CGEQP3 — QR factorization with column pivoting, using Level 3 BLAS. She's the main character, she picks her own columns. 💅

🔥 Purpose

CGEQP3 computes a QR factorization with column pivoting of a matrix A:

A*P = Q*R  using Level 3 BLAS.

🌸 Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N).
- On entry, the M-by-N matrix A.
- On exit, the upper triangle of the array contains the min(M,N)-by-N upper trapezoidal matrix R; the elements below the diagonal, together with the array TAU, represent the unitary matrix Q as a product of min(M,N) elementary reflectors.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
JPVT — JPVT is INTEGER array, dimension (N).
- On entry, if JPVT(J).ne.0, the J-th column of A is permuted to the front of A*P (a leading column); if JPVT(J)=0, the J-th column of A is a free column.
- On exit, if JPVT(J)=K, then the J-th column of A*P was the the K-th column of A.
TAU
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors, periodt. ✨
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO=0, WORK(1) returns the optimal LWORK. The way it just hands you the tea on the way out 💅
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= N+1. For optimal performance LWORK >= ( N+1 )*NB, where NB is the optimal blocksize. Like, you can go smaller, but bestie, don't.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. It's giving "just tell me what you want and I'll handle it" 💖

RWORK — RWORK is REAL array, dimension (2*N)
INFO — INFO is INTEGER.
- = 0: successful exit. slay 🎀
- < 0: if INFO = -i, the i-th argument had an illegal value. not the vibe 😭

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 Further Details:

The matrix Q is represented as a product of elementary reflectors

   Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

🦋 Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
X. Sun, Computer Science Dept., Duke University, USA

🌈 subroutine `cgeqr2` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

   subroutine cgeqr2 (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
       integer INFO)

CGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm. It's the little one that does the whole thing by hand, no blocking, no shortcuts 🫶

Purpose:

CGEQR2 computes a QR factorization of a complex m-by-n matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix;
0 is a (m-n)-by-n zero matrix, if m > n.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the m by n matrix A. On exit, the elements on and above the diagonal of the array contain the min(m,n) by n upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). She stores everything she learned right there in the same array, iconic 💅
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is COMPLEX array, dimension (N)
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 Further Details:

The matrix Q is represented as a product of elementary reflectors

   Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

🦋 subroutine `cgeqr2p` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

   subroutine cgeqr2p (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
       integer INFO)

CGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm. Same energy as cgeqr2, but she's committed to keeping those diagonal entries real and nonnegative 💖 the p stands for "positive, bestie".

Purpose:

CGEQR2P computes a QR factorization of a complex m-by-n matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix with nonnegative diagonal entries;
0 is a (m-n)-by-n zero matrix, if m > n.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the m by n matrix A. On exit, the elements on and above the diagonal of the array contain the min(m,n) by n upper trapezoidal matrix R (R is upper triangular if m >= n). The diagonal entries of R are real and nonnegative; the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). Yeah, she forces the diagonal to behave 🌸
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is COMPLEX array, dimension (N)
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 Further Details:

The matrix Q is represented as a product of elementary reflectors

   Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

See Lapack Working Note 203 for details. 📓 the receipts are there, iykyk

🌈 subroutine `cgeqrf` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

   subroutine cgeqrf (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
       integer LWORK, integer INFO)

CGEQRF — the main character 👑 the blocked one, the one everybody actually calls.

Purpose:

CGEQRF computes a QR factorization of a complex M-by-N matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of min(m,n) elementary reflectors (see Further Details).
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
TAU — TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. Just ask her how much space she needs, she'll tell you 💖

INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💜 Further Details:

The matrix Q is represented as a product of elementary reflectors

   Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

   H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

🦋 subroutine `cgeqrfp` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

   subroutine cgeqrfp (integer M, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
       integer LWORK, integer INFO)

CGEQRFP — blocked edition, but with that nonnegative-diagonal glow-up ✨

Purpose:

CGEQR2P computes a QR factorization of a complex M-by-N matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix with nonnegative diagonal entries;
0 is a (M-N)-by-N zero matrix, if M > N.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n). The diagonal entries of R are real and nonnegative; the elements below the diagonal, with the array TAU, represent the unitary matrix Q as a product of min(m,n) elementary reflectors (see Further Details).
LDA
LDA — INTEGER energy, bestie. It's the leading dimension of the array A. And the rule is LDA >= max(1,M). She's not flexible on this one, it's giving hard boundary. 💅
TAU — TAU is a COMPLEX array, dimension (min(M,N)). It's holding the scalar factors of the elementary reflectors (peep Further Details below for the whole lore 🫶).
WORK — WORK is a COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. like she's literally telling u what she wants upfront, iconic behavior ✨
LWORK — LWORK is INTEGER, the dimension of the array WORK. So: LWORK >= max(1,N). For optimum performance, LWORK >= N*NB, where NB is the optimal blocksize. And here's the special-case tea ☕ — if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. It's giving "just asking, not committing" fr fr.
INFO — INFO is INTEGER.
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 💀

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🦋 Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

See Lapack Working Note 203 for details

🌸 subroutine cgeqrt

subroutine cgeqrt (integer M, integer N, integer NB, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT,
    complex, dimension( * ) WORK, integer INFO)

CGEQRT — say hi to her 💖

Purpose

CGEQRT computes a blocked QR factorization of a complex M-by-N matrix A using the compact WY representation of Q. blocked. compact. efficient. she ate 🔥

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
NB — NB is INTEGER. The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A — A is a COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal are the columns of V.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is a COMPLEX array, dimension (LDT,MIN(M,N)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details 👀
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
WORK — WORK is a COMPLEX array, dimension (NB*N).
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details

The matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A. (they're implied, main character energy, they don't need to announce themselves 💅)

Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each block is of order NB except for the last block, which is of order IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB for the last block) T's are stored in the NB-by-K matrix T as

             T = (T1 T2 ... TB).

🎀 subroutine cgeqrt2

subroutine cgeqrt2 (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( ldt, * ) T, integer LDT, integer INFO)

CGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose

CGEQRT2 computes a QR factorization of a complex M-by-N matrix A, using the compact WY representation of Q.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= N.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is a COMPLEX array, dimension (LDA,N). On entry, the complex M-by-N matrix A. On exit, the elements on and above the diagonal contain the N-by-N upper triangular matrix R; the elements below the diagonal are the columns of V. See below for further details.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is a COMPLEX array, dimension (LDT,N). The N-by-N upper triangular factor of the block reflector. The elements on and above the diagonal contain the block reflector T; the elements below the diagonal are not used. See below for further details.
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
INFO — INFO is INTEGER.
- = 0: successful exit 🫶
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details

The matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A. The block reflector H is then given by

             H = I - V * T * V**H

where V**H is the conjugate transpose of V.

💜 recursive subroutine cgeqrt3

recursive subroutine cgeqrt3 (integer M, integer N, complex, dimension( lda,
    * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, integer
    INFO)

CGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. yes she recurses. she's that girl 🌈

Purpose

CGEQRT3 recursively computes a QR factorization of a complex M-by-N matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M — M is INTEGER. The number of rows of the matrix A. M >= N.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is a COMPLEX array, dimension (LDA,N). On entry, the complex M-by-N matrix A. On exit, the elements on and above the diagonal contain the N-by-N upper triangular matrix R; the elements below the diagonal are the columns of V. See below for further details.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is a COMPLEX array, dimension (LDT,N). The N-by-N upper triangular factor of the block reflector. The elements on and above the diagonal contain the block reflector T; the elements below the diagonal are not used. See below for further details.
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
INFO — INFO is INTEGER.
- = 0: successful exit ⭐
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details

The matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A. The block reflector H is then given by

             H = I - V * T * V**H

where V**H is the conjugate transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

🧚 subroutine cgerfs

subroutine cgerfs (character TRANS, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF,
    integer LDAF, integer, dimension( * ) IPIV, complex, dimension( ldb, * )
    B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real,
    dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * )
    WORK, real, dimension( * ) RWORK, integer INFO)

CGERFS

Purpose

CGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. she's the quality-control bestie of linear systems 💅✨

Parameters

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
N — N is INTEGER. The order of the matrix A. N >= 0.
NRHS —
NRHS — INTEGER, bestie. It's the number of right hand sides, aka how many columns live in the matrices B and X. Rule: NRHS >= 0. 💅
A — COMPLEX array, dimension (LDA,N). It's the original N-by-N matrix A. The og. The blueprint. 🎀
LDA — INTEGER, the leading dimension of the array A. Requirement: LDA >= max(1,N). Respect the dimensions or it's giving chaos. 😭
AF — COMPLEX array, dimension (LDAF,N). These are the factors L and U from the factorization A = P*L*U, exactly as computed by CGETRF. ✨
LDAF — INTEGER, leading dimension of the array AF. LDAF >= max(1,N). Periodt.
IPIV — INTEGER array, dimension (N). The pivot indices from CGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). That's the receipts of every swap. 🔄
B — COMPLEX array, dimension (LDB,NRHS). The right hand side matrix B. 🫶
LDB — INTEGER, leading dimension of the array B. LDB >= max(1,N). No shortcuts, queen. 💖
X — COMPLEX array, dimension (LDX,NRHS). On entry, it's the solution matrix X, as computed by CGETRS. On exit, it's the improved solution matrix X. Glow up moment. 👑
LDX — INTEGER, leading dimension of the array X. LDX >= max(1,N). 🦋
FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X). So if XTRUE is the true solution corresponding to X(j), then FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). And here's the vibe: this estimate is as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. Lowkey humble, highkey safe. 🌸
BERR — REAL array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) — meaning the smallest relative change in any element of A or B that makes X(j) an exact solution. 💀
WORK — COMPLEX array, dimension (2*N). 🧚
RWORK — REAL array, dimension (N). 💜
INFO — INTEGER. The vibe report:
- = 0: successful exit, we ate 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. Not the moment. 🙈

🧠 Internal Parameters: 🧠

ITMAX is the maximum number of steps of iterative refinement. That's it, that's the whole secret sauce. ✨

✍️ Author ✍️

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 subroutine cgerfsx 🌈

subroutine cgerfsx (character TRANS, character EQUED, integer N, integer
    NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension(
    ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real,
    dimension( * ) R, real, dimension( * ) C, complex, dimension( ldb, * )
    B, integer LDB, complex, dimension( ldx , * ) X, integer LDX, real
    RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension(
    nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP,
    integer NPARAMS, real, dimension( * ) PARAMS, complex, dimension( * )
    WORK, real, dimension( * ) RWORK, integer INFO)

CGERFSX

🎯 Purpose: 🎯

CGERFSX improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. In addition to the normwise error bound, the code provides a maximum componentwise error bound if possible. Peep the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those error bounds. 🔥

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED, R and C below. In that case, the solution and error bounds returned are for the original unequilibrated system. She remembers where you came from. 💖

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. And if the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. Iconic, honestly. 🎀

🛠️ Parameters 🛠️

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. No guessing allowed. 💅
- = 'N': No equilibration
- = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
- = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
- = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C). The right hand side B has been changed accordingly.
N — N is INTEGER. The order of the matrix A. N >= 0. 🦋
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 🌸
A — A is COMPLEX array, dimension (LDA,N). The original N-by-N matrix A. ✨
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💖
AF — AF is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF. 🎀
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). 👑
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from CGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). 🔄
R — R is REAL array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed.

And listen, bestie, this part matters: if R is accessed, each element of R should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 💀

C — C is REAL array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed.

Same energy as R, fr fr: if C is accessed, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 😭

B — B is COMPLEX array, dimension (LDB,NRHS). The right hand side matrix B. 🫶
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). 💜
X — X is COMPLEX array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by CGETRS. On exit, the improved solution matrix X. Main character arc. 🌟
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N). 🧚
RCOND — RCOND is REAL. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. So like… don't panic-panic. 🌈
BERR — BERR is REAL array, dimension (NRHS). Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). 🔥
N_ERR_BNDS — N_ERR_BNDS is INTEGER. Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. 👀
ERR_BNDS_NORM — ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

      max_j (abs(XTRUE(j,i) - X(j,i)))
     ------------------------------
           max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.

err = 3 — Reciprocal condition number: the estimated normwise reciprocal condition number. You compare it against the threshold sqrt(n) * slamch('Epsilon') to decide whether the error estimate counts as 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. 🧮✨

👉 Want the deep lore and the extra cautions? Go read Lapack Working Note 165. Bestie homework, but worth it.

🎀 `ERR_BNDS_COMP` 🎀

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array is literally holding all the info about various error bounds and condition numbers that correspond to the componentwise relative error, which is defined like this:

Componentwise relative error in the ith solution vector:
               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (the componentwise relative error depends on it, so yeah) and by the type of error information described below. Right now there are up to three pieces of info returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed — like, not even glanced at. And if N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

err = 1 — the 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). 🫶
err = 2 — the 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). Only trust this error bound if the previous boolean is true, okay? 🙈
err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. 💅

👉 And again: see Lapack Working Note 165 for further details and extra cautions.

✨ `NPARAMS` ✨

NPARAMS is INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. No cap, we just use the defaults. 🧚

✨ `PARAMS` ✨

PARAMS is REAL array, dimension NPARAMS. Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not.
- Default: 1.0
- = 0.0: No refinement is performed, and no error bounds are computed.
- = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
- (other values are reserved for future use) — so don't get creative. 🔒
PARAMS(LA_LINRX_ITHRESH_I = 2) — maximum number of residual computations allowed for refinement.
- Default: 10
- Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. ⚠️
PARAMS(LA_LINRX_CWISE_I = 3) — flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (attempt componentwise convergence)

✨ `WORK` ✨

WORK is COMPLEX array, dimension (2*N).

✨ `RWORK` ✨

RWORK is REAL array, dimension (2*N).

✨ `INFO` ✨

INFO is INTEGER.

= 0: Successful exit. The solution to every right-hand side is guaranteed. 🎉
< 0: If INFO = -i, the i-th argument had an illegal value.
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
= N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 📋

✍️ Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 `subroutine cgerq2` 🌈

subroutine cgerq2 (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer INFO)

CGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm. 🦋

🎯 Purpose:

CGERQ2 computes an RQ factorization of a complex m by n matrix A: A = R * Q.

🎀 Parameters

💜 `M`

M is INTEGER. The number of rows of the matrix A. M >= 0.

💜 `N`

N is INTEGER. The number of columns of the matrix A. N >= 0.

💜 `A`

A is COMPLEX array, dimension (LDA,N). On entry, the m by n matrix A. On exit, if m <= n, the upper triangle of the subarray A(1:m,n-m+1:n) contains the m by m upper triangular matrix R; if m >= n, the elements on and above the (m-n)-th subdiagonal contain the m by n upper trapezoidal matrix R; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). 🪄

💜 `LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

💜 `TAU`

TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).

💜 `WORK`

WORK is COMPLEX array, dimension (M).

💜 `INFO`

INFO is INTEGER.

= 0: successful exit ✨
< 0: if INFO = -i, the i-th argument had an illegal value

✍️ Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

📖 Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i). 🧠💅

🌈 `subroutine cgerqf` 🌈

subroutine cgerqf (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK,
    integer LWORK, integer INFO)

CGERQF

🎯 Purpose:

CGERQF computes an RQ factorization of a complex M-by-N matrix A: A = R * Q.

🎀 Parameters

💜 `M`

M is INTEGER. The number of rows of the matrix A. M >= 0.

💜 `N`

N is INTEGER. The number of columns of the matrix A. N >= 0.

💜 `A`

A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, if m <= n, the upper triangle of the subarray A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R; if m >= n, the elements on and above the (m-n)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAU, represent the unitary matrix Q as a product of min(m,n) elementary reflectors (see Further Details). 🌸

💜 `LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

💜 `TAU`

TAU is COMPLEX array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).

💜 `WORK`

WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

💜 `LWORK`

`LWORK` ✨

okay so LWORK is an INTEGER, and it's literally the dimension of the array WORK. The rules, bestie, listen up: LWORK >= 1 if MIN(M,N) = 0, and LWORK >= M otherwise. And if you want it to be fast fast? For optimum performance, LWORK >= M*NB, where NB is the optimal blocksize. 👑

And the plot twist 🫢 — if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So like, nobody screams at you. 💅

`INFO` 🌈

INFO is INTEGER and it's giving report card:

= 0: successful exit 🎉
< 0: if INFO = -i, the i-th argument had an illegal value 😭

Author 🫶

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details: 👑

The matrix Q is represented as a product of elementary reflectors

Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i). 🦋

👑 `CGESVJ` 👑

subroutine cgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV,
    integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real,
    dimension( n ) SVA, integer MV, complex, dimension( ldv, * ) V, integer
    LDV, complex, dimension( lwork ) CWORK, integer LWORK, real, dimension(
    lrwork ) RWORK, integer LRWORK, integer INFO)

Purpose 💫

CGESVJ computes the singular value decomposition (SVD) of a complex M-by-N matrix A, where M >= N. The SVD of A is written as

                                          [++]   [xx]   [x0]   [xx]
                    A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
                                          [++]   [xx]

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N unitary matrix. The diagonal elements of SIGMA are the singular values of A. The columns of U and V are the left and the right singular vectors of A, respectively. 💜

Parameters 🎀

`JOBA` 🌸

JOBA is CHARACTER*1, and it specifies the structure of A:

'L': the input matrix A is lower triangular;
'U': the input matrix A is upper triangular;
'G': the input matrix A is general M-by-N matrix, M >= N.

`JOBU` 💅

JOBU is CHARACTER*1, and it decides whether to compute the left singular vectors (columns of U):

'U' or 'F': the left singular vectors corresponding to the nonzero singular values are computed and returned in the leading columns of A. See more details in the description of A. The default numerical orthogonality threshold is set to approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
'C': analogous to JOBU='U', except that the user can control the level of numerical orthogonality of the computed left singular vectors. TOL can be set to TOL = CTOL*EPS, where CTOL is given on input in the array WORK. No CTOL smaller than ONE is allowed. CTOL greater than 1 / EPS is meaningless. The option 'C' can be used if M*EPS is satisfactory orthogonality of the computed left singular vectors, so CTOL=M could save a few sweeps of Jacobi rotations. See the descriptions of A and WORK(1).
'N': the matrix U is not computed. However, see the description of A. 🙈

`JOBV` 🦋

JOBV is CHARACTER*1, and it decides whether to compute the right singular vectors, that is, the matrix V:

'V' or 'J': the matrix V is computed and returned in the array V
'A': the Jacobi rotations are applied to the MV-by-N array V. In other words, the right singular vector matrix V is not computed explicitly; instead it is applied to an MV-by-N matrix initially stored in the first MV rows of V.
'N': the matrix V is not computed and the array V is not referenced

`M` 🌈

M is INTEGER. The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0.

`N` ⭐

N is INTEGER. The number of columns of the input matrix A. M >= N >= 0.

`A` 💖

A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit:

If JOBU = 'U' .OR. JOBU = 'C':
- If INFO = 0: RANKA orthonormal columns of U are returned in the leading RANKA columns of the array A. Here RANKA <= N is the number of computed singular values of A that are above the underflow threshold SLAMCH('S'). The singular vectors corresponding to underflowed or zero singular values are not computed. The value of RANKA is returned in the array RWORK as RANKA=NINT(RWORK(2)). Also see the descriptions of SVA and RWORK. The computed columns of U are mutually numerically orthogonal up to approximately TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'), see the description of JOBU.
- If INFO > 0: the procedure CGESVJ did not converge in the given number of iterations (sweeps). In that case, the computed columns of U may not be orthogonal up to TOL. The output U (stored in A), SIGMA (given by the computed singular values in SVA(1:N)) and V is still a decomposition of the input matrix A in the sense that the residual || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small. 😭
If JOBU = 'N':
- If INFO = 0: note that the left singular vectors are 'for free' in the one-sided Jacobi SVD algorithm. However, if only the singular values are needed, the level of numerical orthogonality of U is not an issue and iterations are stopped when the columns of the iterated matrix are numerically orthogonal up to approximately M*EPS. Thus, on exit, A contains the columns of U scaled with the corresponding singular values.
- If INFO > 0: the procedure CGESVJ did not converge in the given number of iterations (sweeps).

`LDA` 🎀

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

`SVA` 🌸

SVA is REAL array, dimension (N). On exit:

If INFO = 0: depending on the value SCALE = RWORK(1), we have:

If SCALE = ONE: SVA(1:N) contains the computed singular values of A. During the computation SVA contains the Euclidean column norms of the iterated matrices in the array A.
If SCALE .NE. ONE: the singular values of A are SCALE*SVA(1:N), and this factored representation is due to the fact that some of the singular values of A might underflow or overflow. 😱

If INFO > 0: the procedure CGESVJ did not converge in the given number of iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. 💀

`MV` 🫶

MV is INTEGER. If JOBV = 'A', then the product of Jacobi rotations in CGESVJ is applied to the first MV rows of V. See the description of JOBV.

`V` 💜

V is COMPLEX array, dimension (LDV,N).

If JOBV = 'V', then V contains on exit the N-by-N matrix of the right singular vectors;
If JOBV = 'A', then V contains the product of the computed right singular vector matrix and the initial matrix in the array V.
If JOBV = 'N', then V is not referenced.

`LDV` ✨

LDV is INTEGER. The leading dimension of the array V, LDV >= 1.

If JOBV = 'V', then LDV >= max(1,N).
If JOBV = 'A', then LDV >= max(1,MV).

`CWORK` 🍓

CWORK is COMPLEX array, dimension (max(1,LWORK)). Used as workspace.

If on entry LWORK = -1, then a workspace query is assumed and no computation is done; CWORK(1) is set to the minimal (and optimal) length of CWORK. 🕵️♀️

`LWORK` 🔥

LWORK is INTEGER. Length of CWORK, LWORK >= M+N.

`RWORK` 🧚

RWORK — REAL array, dimension (max(6,LRWORK)). bestie this array is carrying SO much intel, it's literally the group chat receipts of the whole factorization 💖

On entry,

If JOBU = 'C':
- RWORK(1) = CTOL, where CTOL defines the threshold for convergence. The process stops if all columns of A are mutually orthogonal up to CTOL*EPS, EPS=SLAMCH('E'). It is required that CTOL >= ONE, i.e. it is not allowed to force the routine to obtain orthogonality below EPSILON.

On exit,

RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N) are the computed singular values of A. (See description of SVA().)
RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero singular values.
RWORK(3) = NINT(RWORK(3)) is the number of the computed singular values that are larger than the underflow threshold.
RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi rotations needed for numerical convergence.
RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep. This is useful information in cases when CGESVJ did not converge, as it can be used to estimate whether the output is still useful and for post festum analysis.
RWORK(6) = the largest absolute value over all sines of the Jacobi rotation angles in the last sweep. It can be useful for a post festum analysis.

If on entry LRWORK = -1, then a workspace query is assumed and no computation is done; RWORK(1) is set to the minial (and optimal) length of RWORK.

LRWORK — INTEGER. Length of RWORK, LRWORK >= MAX(6,N). no cap, just give it room to breathe 🫶
INFO — INTEGER, the vibe report at the end:
- = 0: successful exit. slay 💅
- < 0: if INFO = -i, then the i-th argument had an illegal value.
- > 0: CGESVJ did not converge in the maximal allowed number (NSWEEP=30) of sweeps. The output may still be useful. See the description of RWORK.

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details: 🔍

The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations. In the case of underflow of the tangent of the Jacobi angle, a modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses column interchanges of de Rijk [1]. The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [2]. The condition number that determines the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [4,5], and it is the kernel routine in the SIGMA library [6]. Some tuning parameters (marked with [TP]) are available for the implementer.

The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits. 😭 iconic behaviour honestly

🌸 Contributor: 🌸

============

Zlatko Drmac (Zagreb, Croatia)

📚 References: 📚

P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer. SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
Z. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic. SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. LAPACK Working note 169.
Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. LAPACK Working note 170.
Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations. Department of Mathematics, University of Zagreb, 2008, 2015.

🐛 Bugs, examples and comments: 🐛

===========================
Please report all bugs and send interesting test examples and comments to
drmac@math.hr. Thank you.

fr fr, if you find a bug, tell the man. he's nice about it. 💜

✨ `subroutine cgetf2 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)` ✨

CGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

🎯 Purpose: 🎯

CGETF2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row interchanges.

The factorization has the form

   A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This is the right-looking Level 2 BLAS version of the algorithm.

🧩 Parameters 🧩

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the m by n matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
INFO — INTEGER:
- = 0: successful exit
- < 0: if INFO = -k, the k-th argument had an illegal value
- > 0: if INFO = k, U(k,k) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀 not a drill bestie

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ `subroutine cgetrf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)` ✨

CGETRF CGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm

CGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

🎯 Purpose: 🎯

CGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

   A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This is the right-looking Level 3 BLAS version of the algorithm.

🧩 Parameters 🧩

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
INFO — INTEGER:
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎯 Purpose: 🎯

CGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

   A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

🧩 Parameters 🧩

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
INFO —
INFO — INTEGER, bestie, and this one is the receipts ✨

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, U(i,i) is exactly zero. The factorization
      has been completed, but the factor U is exactly
      singular, and division by zero will occur if it is used
      to solve a system of equations.

Translation: INFO = 0 means we ate. INFO = -i means argument number i was giving illegal-value energy. And INFO = i means U(i,i) is exactly zero — the factorization finished but U is fully singular, so the second you try to solve a system with it you're dividing by zero. 💀

🫶 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

📅 Date

December 2016

🎯 Purpose:

CGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. No cap, that's the whole job.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n). 🦋

This code implements an iterative version of Sivan Toledo's recursive LU algorithm[1]. For square matrices, this iterative versions should be within a factor of two of the optimum number of memory transfers. (yes, "versions", we're keeping it, that's what it says 💅)

The pattern is as follows, with the large blocks of U being updated in one call to DTRSM, and the dotted lines denoting sections that have had all pending permutations applied:

 1 2 3 4 5 6 7 8
+-+-+---+-------+------
| |1|   |       |
|.+-+ 2 |       |
| | |   |       |
|.|.+-+-+   4   |
| | | |1|       |
| | |.+-+       |
| | | | |       |
|.|.|.|.+-+-+---+  8
| | | | | |1|   |
| | | | |.+-+ 2 |
| | | | | | |   |
| | | | |.|.+-+-+
| | | | | | | |1|
| | | | | | |.+-+
| | | | | | | | |
|.|.|.|.|.|.|.|.+-----
| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in the binary expansion of the current column. Each Schur update is applied as soon as the necessary portion of U is available. it's giving binary-indexed main character energy fr fr 🌈

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997), 1065-1081. http://dx.doi.org/10.1137/S0895479896297744

🌸 Parameters

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored. (not stored!! don't go looking for them 🙈)
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
INFO — INTEGER.

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, U(i,i) is exactly zero. The factorization
      has been completed, but the factor U is exactly
      singular, and division by zero will occur if it is used
      to solve a system of equations.

🫶 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

📅 Date

December 2016

🔁 `CGETRF2` — the recursive one

recursive subroutine cgetrf2 (integer M, integer N, complex, dimension( lda,
* ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

🎯 Purpose:

CGETRF2 computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This is the recursive version of the algorithm. It divides the matrix into four submatrices:

       [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
   A = [ -----|----- ]  with n1 = min(m,n)/2
       [  A21 | A22  ]       n2 = n-n1

                                      [ A11 ]
The subroutine calls itself to factor [ --- ],
                                      [ A12 ]
                [ A12 ]
do the swaps on [ --- ], solve A12, update A22,
                [ A22 ]

then calls itself to factor A22 and do the swaps on A21. it's recursion bestie, it contains itself, iconic 🪆✨

🌸 Parameters

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
INFO — INTEGER.

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, U(i,i) is exactly zero. The factorization
      has been completed, but the factor U is exactly
      singular, and division by zero will occur if it is used
      to solve a system of equations.

🫶 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔄 `CGETRI` — the un-factorizer

subroutine cgetri (integer N, complex, dimension( lda, * ) A, integer LDA,
integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer
LWORK, integer INFO)

🎯 Purpose:

CGETRI computes the inverse of a matrix using the LU factorization computed by CGETRF.

This method inverts U and then computes inv(A) by solving the system inv(A)*L = inv(U) for inv(A). we love a little algebraic sleight of hand 💅

🌸 Parameters

N — INTEGER. The order of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the factors L and U from the factorization A = P*L*U as computed by CGETRF. On exit, if INFO = 0, the inverse of the original matrix A. slay, in-place glow-up ✨
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from CGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,N). For optimal performance LWORK >= N*NB, where NB is the optimal blocksize returned by ILAENV.

INFO — INTEGER.

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
      singular and its inverse could not be computed.

Heads up bestie: if INFO > 0, U(i,i) is exactly zero — the matrix is singular and its inverse could not be computed. No inverse for you, periodt. 😭

🫶 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🧮 `CGETRS` — the solver

subroutine cgetrs (character TRANS, integer N, integer NRHS, complex,
dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
complex, dimension( ldb, * ) B, integer LDB, integer INFO)

🎯 Purpose:

CGETRS solves a system of linear equations

   A * X = B,  A**T * X = B,  or  A**H * X = B

with a general N-by-N matrix A using the LU factorization computed by CGETRF. plug in the factorization, get out the answer 🫶

🌸 Parameters

TRANS — CHARACTER*1. Specifies the form of the system of equations:

= 'N':  A * X = B     (No transpose)
= 'T':  A**T * X = B  (Transpose)
= 'C':  A**H * X = B  (Conjugate transpose)

N — INTEGER. The order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A — COMPLEX array, dimension (LDA,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from CGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
B — COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. in goes B, out comes X, a true transformation arc 🌈
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INTEGER.

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value

🫶 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

👑 `CHGEQZ`

subroutine chgeqz (character JOB, character COMPQ, character COMPZ, integer
N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer
LDH, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( *
) ALPHA, complex, dimension( * ) BETA, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer
INFO)

🎯 Purpose:

WORK — a COMPLEX array, dimension (MAX(1,LWORK)). On exit, as long as INFO >= 0, then WORK(1) hands you back the optimal LWORK. cute, right? 💖
LWORK — an INTEGER, aka how big the WORK array is. LWORK >= max(1,N).

ok but here's the fun part bestie: if LWORK = -1, it's a workspace query ✨. The routine does NOT do the actual work — it just figures out the optimal size of WORK, sticks that number in the first entry of WORK, and XERBLA will not come for you about LWORK. no error message, no drama. 🫶

RWORK — a REAL array, dimension (N).
INFO — an INTEGER, and this one is the vibe check on whether anything went wrong 👀
- = 0: successful exit 🎉 slay.
- < 0: if INFO = -i, the i-th argument had an illegal value. you passed something cursed. 😭
- = 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, BUT ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.
- = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N should be correct.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 Further Details: 🌈

We assume that complex ABS works as long as its value is less than overflow. that's the whole assumption, that's the whole vibe, periodt. 💅

🦋 subroutine cla_geamv 🦋

subroutine cla_geamv (integer TRANS, integer M, integer N, real ALPHA,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X,
    integer INCX, real BETA, real, dimension( * ) Y, integer INCY)

CLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds. yes she's the error-bounds girlie. 💖

✨ Purpose ✨

ok so CLA_GEAMV performs one of the matrix-vector operations

        y := alpha*abs(A)*abs(x) + beta*abs(y),
   or   y := alpha*abs(A)**T*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an m by n matrix. no cap.

This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. A zero entry is considered 'symbolic' if all multiplications involved in computing that entry have at least one zero multiplicand. iykyk 🧚

🎀 Parameters 🎀

TRANS — an INTEGER. On entry, TRANS specifies the operation to be performed as follows:

  BLAS_NO_TRANS      y := alpha*abs(A)*abs(x) + beta*abs(y)
  BLAS_TRANS         y := alpha*abs(A**T)*abs(x) + beta*abs(y)
  BLAS_CONJ_TRANS    y := alpha*abs(A**T)*abs(x) + beta*abs(y)

Unchanged on exit. (yes, BLAS_TRANS and BLAS_CONJ_TRANS are giving the exact same formula up there — we do not editorialize, we just report. 💀)

M — an INTEGER. On entry, M specifies the number of rows of the matrix A. M must be at least zero. Unchanged on exit.
N — an INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.
ALPHA — a REAL. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
A — a COMPLEX array, dimension (LDA,n). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit.
LDA — an INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ). Unchanged on exit.
X — a COMPLEX array, dimension ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. Before entry, the incremented array X must contain the vector x. Unchanged on exit.
INCX
INCX — INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. Lowkey important: INCX must not be zero. 🚫 Unchanged on exit.
BETA — BETA is REAL. On entry, BETA specifies the scalar beta, bestie. When BETA is supplied as zero then Y need not be set on input. (Iconic little shortcut tbh 💅) Unchanged on exit.
Y — Y is REAL array, dimension ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. ✨ She gets a whole glow-up on the way out.
INCY — INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

Level 2 Blas routine.

Author 🫶

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

real function `cla_gercond_c`

real function cla_gercond_c (character TRANS, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    integer, dimension( * ) IPIV, real, dimension( * ) C, logical CAPPLY,
    integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices. Slay.

Purpose:

CLA_GERCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector. That's the whole vibe. 💖

Parameters

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate Transpose = Transpose)
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by CGETRF; row i of the matrix was interchanged with row IPIV(i). 🎀
C — C is REAL array, dimension (N). The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — CAPPLY is LOGICAL. If .TRUE. then access the vector C in the formula above.
INFO — INFO is INTEGER.
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

real function `cla_gercond_x`

real function cla_gercond_x (character TRANS, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    integer, dimension( * ) IPIV, complex, dimension( * ) X, integer INFO,
    complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices. 💅

Purpose:

CLA_GERCOND_X computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX vector. Note the glow-up from the last one: diag(X) not inv(diag(C)). 👑

Parameters

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate Transpose = Transpose)
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by CGETRF; row i of the matrix was interchanged with row IPIV(i). 🦋
X — X is COMPLEX array, dimension (N). The vector X in the formula op(A) * diag(X).
INFO — INFO is INTEGER.
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

subroutine `cla_gerfsx_extended`

subroutine cla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE,
    integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA,
    complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * )
    IPIV, logical COLEQU, real, dimension( * ) C, complex, dimension( ldb, *
    ) B, integer LDB, complex, dimension( ldy, * ) Y, integer LDY, real,
    dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * )
    ERRS_N, real, dimension( nrhs, * ) ERRS_C, complex, dimension( * ) RES,
    real, dimension( * ) AYB, complex, dimension( * ) DY, complex,
    dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real
    DZ_UB, logical IGNORE_CWISE, integer INFO)

CLA_GERFSX_EXTENDED

Purpose:

CLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by CGERFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERRS_N and ERRS_C for details of the error bounds. Note that this subroutine is only responsible for setting the second fields of ERRS_N and ERRS_C. So like… she's not doing the whole job, she's doing her part of the job. Teamwork, bestie. 🫶

Parameters

PREC_TYPE — PREC_TYPE is INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P
- = 'S': Single
- = 'D': Double
- = 'I': Indigenous
- = 'X' or 'E': Extra
TRANS_TYPE — TRANS_TYPE is INTEGER. Specifies the transposition operation on A. The value is defined by ILATRANS(T) where T is a CHARACTER and T
- = 'N': No transpose
- = 'T': Transpose
- = 'C': Conjugate transpose
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — NRHS is INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by CGETRF; row i of the matrix was interchanged with row IPIV(i). 🍓
COLEQU — COLEQU is LOGICAL. If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
C — C is REAL array, dimension (N). The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. ⚠️ Like, read that twice, bestie.
B — B is COMPLEX array, dimension (LDB,NRHS). The right-hand-side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Y — Y is COMPLEX array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by CGETRS. On exit, the improved solution matrix Y. ✨ glow-up achieved ✨
LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT — BERR_OUT is REAL array, dimension (NRHS). On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula

    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by CLA_LIN_BERR.

N_NORMS — N_NORMS is INTEGER. Determines which error bounds to return (see ERRS_N and ERRS_C). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.
ERRS_N
ERRS_N — ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERRS_N(i,:) corresponds to the ith right-hand side.

The second index in ERRS_N(:,err) contains the following three fields:

err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. 💅

ERRS_C — ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERRS_C(i,:) corresponds to the ith right-hand side.

The second index in ERRS_C(:,err) contains the following three fields:

err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. ✨

RES — RES is COMPLEX array, dimension (N). Workspace to hold the intermediate residual. Slay.
AYB — AYB is REAL array, dimension (N). Workspace.
DY — DY is COMPLEX array, dimension (N). Workspace to hold the intermediate solution.
Y_TAIL — Y_TAIL is COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution.
RCOND — RCOND is REAL. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. It's giving "don't panic but also maybe panic." 😭
ITHRESH — ITHRESH is INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERRS_N and ERRS_C may no longer be trustworthy. Red flag, bestie. 🚩
RTHRESH — RTHRESH is REAL. Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.
DZ_UB — DZ_UB is REAL. Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.
IGNORE_CWISE — IGNORE_CWISE is LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. She said "not today." 🙈
INFO — INFO is INTEGER. = 0: Successful exit. < 0: if INFO = -i, the ith argument to CGETRS had an illegal value. We love a clean exit. 👑

💖 Author 💖

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

✨ real function `cla_gerpvgrw` ✨

real function cla_gerpvgrw (integer N, integer NCOLS, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF)

CLA_GERPVGRW multiplies a square real matrix by a complex matrix.

Purpose:

CLA_GERPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. So like… maybe don't trust it if it's giving tiny numbers. 😬

🌸 Parameters 🌸

N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NCOLS — NCOLS is INTEGER. The number of columns of the matrix A. NCOLS >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).

💖 Author 💖

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 recursive subroutine `claqz0` 🌈

recursive subroutine claqz0 (character, intent(in) WANTS, character,
       intent(in) WANTQ, character, intent(in) WANTZ, integer, intent(in) N,
       integer, intent(in) ILO, integer, intent(in) IHI, complex, dimension(
       lda, * ), intent(inout) A, integer, intent(in) LDA, complex, dimension(
       ldb, * ), intent(inout) B, integer, intent(in) LDB, complex, dimension(
       * ), intent(inout) ALPHA, complex, dimension( * ), intent(inout) BETA,
       complex, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ,
       complex, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ,
       complex, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK,
       real, dimension( * ), intent(out) RWORK, integer, intent(in) REC,
       integer, intent(out) INFO)

CLAQZ0

Purpose:

CLAQZ0 computes the eigenvalues of a matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a matrix pair (A,B):

   A = Q1*H*Z1**H,  B = Q1*T*Z1**H,

as computed by CGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form,

   H = Q*S*Z**H,  T = Q*P*Z**H,

where Q and Z are unitary matrices, P and S are an upper triangular matrices. Iconic. 🦋

Bestie, listen — you can level up your Schur factors if you want. 💅

Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta), where alpha is complex and beta real. If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)

A*x = lambda*B*x

and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP

mu*A*y = B*y.

Eigenvalues can be read directly from the generalized Schur form — literally just peek at the diagonal, no cap:

alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973), pp. 241--256.

Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ Algorithm with Aggressive Early Deflation', SIAM J. Numer. Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift, multipole rational QZ method with agressive early deflation'

🎀 Parameters 🎀

WANTS — WANTS is CHARACTER*1
- = 'E': Compute eigenvalues only;
- = 'S': Compute eigenvalues and the Schur form.
WANTQ — WANTQ is CHARACTER*1
- = 'N': Left Schur vectors (Q) are not computed;
- = 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (A,B) is returned;
- = 'V': Q must contain an unitary matrix Q1 on entry and the product Q1*Q is returned.
WANTZ — WANTZ is CHARACTER*1
- = 'N': Right Schur vectors (Z) are not computed;
- = 'I': Z is initialized to the unit matrix and the matrix Z of right Schur vectors of (A,B) is returned;
- = 'V': Z must contain an unitary matrix Z1 on entry and the product Z1*Z is returned.
N — N is INTEGER — the order of the matrices A, B, Q, and Z. N >= 0.
ILO — ILO is INTEGER — yeah that's it, it just is one, bestie.
IHI — IHI is INTEGER — ILO and IHI mark the rows and columns of A which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
A — A is COMPLEX array, dimension (LDA, N) — On entry, the N-by-N upper Hessenberg matrix A. On exit, if JOB = 'S', A contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal of A matches that of S, but the rest of A is unspecified.
LDA — LDA is INTEGER — the leading dimension of the array A. LDA >= max( 1, N ).
B — B is COMPLEX array, dimension (LDB, N) — On entry, the N-by-N upper triangular matrix B. On exit, if JOB = 'S', B contains the upper triangular matrix P from the generalized Schur factorization. If JOB = 'E', the diagonal of B matches that of P, but the rest of B is unspecified.
LDB — LDB is INTEGER — the leading dimension of the array B. LDB >= max( 1, N ).
ALPHA — ALPHA is COMPLEX array, dimension (N) — each scalar alpha defining an eigenvalue of GNEP.
BETA — BETA is COMPLEX array, dimension (N) — the scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed. 🙈
Q — Q is COMPLEX array, dimension (LDQ, N) — On entry, if COMPQ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', the unitary matrix of left Schur vectors of (A,B), and if COMPQ = 'V', the unitary matrix of left Schur vectors of (A,B). Not referenced if COMPQ = 'N'.
LDQ — LDQ is INTEGER — the leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N.
Z — Z is COMPLEX array, dimension (LDZ, N) — On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of right Schur vectors of (A,B). Not referenced if COMPZ = 'N'.
LDZ — LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)) — On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER — the dimension of the array WORK. LWORK >= max(1,N).

RWORK — RWORK is REAL array, dimension (N)
REC — REC is INTEGER — REC indicates the current recursion level. Should be set to 0 on first call.
INFO — INFO is INTEGER
- = 0: successful exit 🌈
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1,...,N: the QZ iteration did not converge. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.

👑 Author 👑

Thijs Steel, KU Leuven

📅 Date 📅

May 2020

🦋 subroutine `claqz1` 🦋

subroutine claqz1 (logical, intent(in) ILQ, logical, intent(in) ILZ,
     integer, intent(in) K, integer, intent(in) ISTARTM, integer, intent(in)
     ISTOPM, integer, intent(in) IHI, complex, dimension( lda, * ) A,
     integer, intent(in) LDA, complex, dimension( ldb, * ) B, integer,
     intent(in) LDB, integer, intent(in) NQ, integer, intent(in) QSTART,
     complex, dimension( ldq, * ) Q, integer, intent(in) LDQ, integer,
     intent(in) NZ, integer, intent(in) ZSTART, complex, dimension( ldz, * )
     Z, integer, intent(in) LDZ)
     CLAQZ1

✨ Purpose: ✨

CLAQZ1 chases a 1x1 shift bulge in a matrix pencil down a single position. It's giving herding a tiny bulge one step at a time. 🐛

💖 Parameters 💖

ILQ — ILQ is LOGICAL — Determines whether or not to update the matrix Q.
ILZ — ILZ is LOGICAL — Determines whether or not to update the matrix Z.
K — K is INTEGER — Index indicating the position of the bulge. On entry, the bulge is located in (A(k+1,k),B(k+1,k)). On exit, the bulge is located in (A(k+2,k+1),B(k+2,k+1)).
ISTARTM — ISTARTM is INTEGER
ISTOPM — ISTOPM is INTEGER — Updates to (A,B) are restricted to (istartm:k+2,k:istopm). It is assumed without checking that istartm <= k+1 and k+2 <= istopm.
IHI — IHI is INTEGER
A — A is COMPLEX array, dimension (LDA,N)
LDA — LDA is INTEGER — The leading dimension of A as declared in the calling procedure.
B — B is COMPLEX array, dimension (LDB,N)
LDB — LDB is INTEGER — The leading dimension of B as declared in the calling procedure.
NQ — NQ is INTEGER — The order of the matrix Q.
QSTART — QSTART is INTEGER — Start index of the matrix Q. Rotations are applied to columns k+2-qStart:k+3-qStart of Q.
Q — Q is COMPLEX array, dimension (LDQ,NQ)
LDQ — LDQ is INTEGER — The leading dimension of Q as declared in the calling procedure.
NZ — NZ is INTEGER — The order of the matrix Z.
ZSTART — ZSTART is INTEGER — Start index of the matrix Z. Rotations are applied to columns k+1-qStart:k+2-qStart of Z.
Z — Z is COMPLEX array, dimension (LDZ,NZ)
LDZ — LDZ is INTEGER — The leading dimension of Q as declared in the calling procedure.

👑 Author 👑

Thijs Steel, KU Leuven

📅 Date 📅

May 2020

`claqz2` 💖

Okay so this one is a recursive subroutine and it's giving main character energy from the jump, look at this signature:

   recursive subroutine claqz2 (logical, intent(in) ILSCHUR, logical,
       intent(in) ILQ, logical, intent(in) ILZ, integer, intent(in) N, integer,
       intent(in) ILO, integer, intent(in) IHI, integer, intent(in) NW,
       complex, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA,
       complex, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB,
       complex, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ,
       complex, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ,
       integer, intent(out) NS, integer, intent(out) ND, complex, dimension( *
       ), intent(inout) ALPHA, complex, dimension( * ), intent(inout) BETA,
       complex, dimension( ldqc, * ) QC, integer, intent(in) LDQC, complex,
       dimension( ldzc, * ) ZC, integer, intent(in) LDZC, complex, dimension( *
       ) WORK, integer, intent(in) LWORK, real, dimension( * ) RWORK, integer,
       intent(in) REC, integer, intent(out) INFO)

Purpose ✨

CLAQZ2 performs AED.

That's it. That's the whole purpose. No notes. She performs AED and she's not elaborating 💅

Parameters 🎀

ILSCHUR — ILSCHUR is LOGICAL. Determines whether or not to update the full Schur form.
ILQ — ILQ is LOGICAL. Determines whether or not to update the matrix Q.
ILZ — ILZ is LOGICAL. Determines whether or not to update the matrix Z.
N — N is INTEGER. The order of the matrices A, B, Q, and Z. N >= 0.
ILO — ILO is INTEGER.
IHI — IHI is INTEGER. ILO and IHI mark the rows and columns of (A,B) which are to be normalized.
NW — NW is INTEGER. The desired size of the deflation window.
A — A is COMPLEX array, dimension (LDA, N).
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max( 1, N ).
B — B is COMPLEX array, dimension (LDB, N).
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max( 1, N ).
Q — Q is COMPLEX array, dimension (LDQ, N).
LDQ — LDQ is INTEGER.
Z — Z is COMPLEX array, dimension (LDZ, N).
LDZ — LDZ is INTEGER.
NS — NS is INTEGER. The number of unconverged eigenvalues available to use as shifts.
ND — ND is INTEGER. The number of converged eigenvalues found.
ALPHA — ALPHA is COMPLEX array, dimension (N). Each scalar alpha defining an eigenvalue of GNEP.
BETA — BETA is COMPLEX array, dimension (N). The scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.
QC — QC is COMPLEX array, dimension (LDQC, NW).
LDQC — LDQC is INTEGER.
ZC — ZC is COMPLEX array, dimension (LDZC, NW).
LDZC — LDZ is INTEGER. (yes bestie, it says LDZ right there in the LDZC slot — we are simply the messenger, we do not edit 💀)
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N).

RWORK — RWORK is REAL array, dimension (N).
REC — REC is INTEGER. REC indicates the current recursion level. Should be set to 0 on first call.
\param[out] INFO

\verbatim
         INFO is INTEGER
         = 0: successful exit
         < 0: if INFO = -i, the i-th argument had an illegal value

Author 👑

Thijs Steel, KU Leuven, KU Leuven (twice, because once was simply not enough, we love a double-KU-Leuven moment)

Date 📅

May 2020

`claqz3` 🦋

Next up, another subroutine, and this one's whole thing is one single multishift QZ sweep. Signature time:

   subroutine claqz3 (logical, intent(in) ILSCHUR, logical, intent(in) ILQ,
       logical, intent(in) ILZ, integer, intent(in) N, integer, intent(in) ILO,
       integer, intent(in) IHI, integer, intent(in) NSHIFTS, integer,
       intent(in) NBLOCK_DESIRED, complex, dimension( * ), intent(inout) ALPHA,
       complex, dimension( * ), intent(inout) BETA, complex, dimension( lda, *
       ), intent(inout) A, integer, intent(in) LDA, complex, dimension( ldb, *
       ), intent(inout) B, integer, intent(in) LDB, complex, dimension( ldq, *
       ), intent(inout) Q, integer, intent(in) LDQ, complex, dimension( ldz, *
       ), intent(inout) Z, integer, intent(in) LDZ, complex, dimension( ldqc, *
       ), intent(inout) QC, integer, intent(in) LDQC, complex, dimension( ldzc,
       * ), intent(inout) ZC, integer, intent(in) LDZC, complex, dimension( *
       ), intent(inout) WORK, integer, intent(in) LWORK, integer, intent(out)
       INFO)

Purpose ✨

CLAQZ3 Executes a single multishift QZ sweep. One sweep. Multishift. Iconic behavior, no cap.

Parameters 🎀

ILSCHUR — ILSCHUR is LOGICAL. Determines whether or not to update the full Schur form.
ILQ — ILQ is LOGICAL. Determines whether or not to update the matrix Q.
ILZ — ILZ is LOGICAL. Determines whether or not to update the matrix Z.
N — N is INTEGER. The order of the matrices A, B, Q, and Z. N >= 0.
ILO — ILO is INTEGER.
IHI — IHI is INTEGER.
NSHIFTS — NSHIFTS is INTEGER. The desired number of shifts to use.
NBLOCK_DESIRED — NBLOCK_DESIRED is INTEGER. The desired size of the computational windows.
ALPHA — ALPHA is COMPLEX array. SR contains the alpha parts of the shifts to use.
BETA — BETA is COMPLEX array. SS contains the scale of the shifts to use.
A — A is COMPLEX array, dimension (LDA, N).
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max( 1, N ).
B — B is COMPLEX array, dimension (LDB, N).
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max( 1, N ).
Q — Q is COMPLEX array, dimension (LDQ, N).
LDQ — LDQ is INTEGER.
Z — Z is COMPLEX array, dimension (LDZ, N).
LDZ — LDZ is INTEGER.
QC — QC is COMPLEX array, dimension (LDQC, NBLOCK_DESIRED).
LDQC — LDQC is INTEGER.
ZC — ZC is COMPLEX array, dimension (LDZC, NBLOCK_DESIRED).
LDZC — LDZ is INTEGER. (again!! same plot twist as before 😭)
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N).

INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

Author 👑

Thijs Steel, KU Leuven

Date 📅

May 2020

`claunhr_col_getrfnp` 🌸

Okay this one's name is a whole mouthful but she's important, she's the auxiliary bestie doing the modified LU factorization with no pivoting. Here's the fit:

   subroutine claunhr_col_getrfnp (integer M, integer N, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( * ) D, integer INFO)

Purpose ✨

CLAUNHR_COL_GETRFNP computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form:

A - S = L * U,

where:

S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of 'modified' Gaussian elimination is at least one in absolute value (so that division-by-zero not not possible during the division by the diagonal element); (yes, the manpage really does say "not not" — we're quoting, we're not editing, bestie 🙈)
L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1].

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine CLAUNHR_COL_GETRFNP2.

[1] 'Reconstructing Householder vectors from tall-skinny QR', G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

Parameters 🎀

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A-S=L*U; the unit diagonal elements of L are not stored.
LDA
LDA — INTEGER, and it's the leading dimension of the array A, bestie. The rule: LDA >= max(1,M). That's the whole thing, no cap. 🎀
D — COMPLEX array, dimension min(M,N). These are the diagonal elements of the diagonal M-by-N sign matrix S, so D(i) = S(i,i), where 1 <= i <= min(M,N). And the elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ) — like, there are exactly two options and that's the entire menu. 💅
INFO — INTEGER, the little status report we all wait for:
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

🫶 Author 🫶

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌸 Contributors: 🌸

November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley

🧚 `claunhr_col_getrfnp2` 🧚

recursive subroutine claunhr_col_getrfnp2 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) D, integer INFO)

CLAUNHR_COL_GETRFNP2 — say that five times fast, I dare you. 🦋

✨ Purpose ✨

Okay so CLAUNHR_COL_GETRFNP2 computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form:

    A - S = L * U,

where:

S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of 'modified' Gaussian elimination is at least one in absolute value (so that division-by-zero not possible during the division by the diagonal element); 💀 (yes, that's the whole point, no division by zero today)
L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. Iconic behaviour honestly. 👑

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the recursive version of the LU factorization algorithm. Denote A - S by B. The algorithm divides the matrix B into four submatrices:

       [  B11 | B12  ]  where B11 is n1 by n1,
   B = [ -----|----- ]        B21 is (m-n1) by n1,
       [  B21 | B22  ]        B12 is n1 by n2,
                              B22 is (m-n1) by n2,
                              with n1 = min(m,n)/2, n2 = n-n1.

The subroutine calls itself to factor B11, solves for B21, solves for B12, updates B22, then calls itself to factor B22. Recursion said "I'll do it myself" and honestly? She ate. 🔥

For more details on the recursive LU algorithm, see [2].

CLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked routine CLAUNHR_COL_GETRFNP, which uses blocked code calling Level 3 BLAS to update the submatrix. However, CLAUNHR_COL_GETRFNP2 is self-sufficient and can be used without CLAUNHR_COL_GETRFNP. Independent queen energy, periodt. 💜

[1] 'Reconstructing Householder vectors from tall-skinny QR', G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.
[2] 'Recursion leads to automatic variable blocking for dense linear algebra algorithms', F. Gustavson, IBM J. of Res. and Dev., vol. 41, no. 6, pp. 737-755, 1997.

📋 Parameters 📋

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A-S=L*U; the unit diagonal elements of L are not stored.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
D — COMPLEX array, dimension min(M,N). The diagonal elements of the diagonal M-by-N sign matrix S, D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
INFO — INTEGER:
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

🫶 Author 🫶

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌸 Contributors: 🌸

November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley

🦋 `ctgevc` 🦋

subroutine ctgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( lds, * ) S, integer LDS, complex, dimension( ldp, * ) P, integer LDP, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTGEVC — get those eigenvectors, bestie. 🎀

✨ Purpose ✨

CTGEVC computes some or all of the right and/or left eigenvectors of a pair of complex matrices (S,P), where S and P are upper triangular. Matrix pairs of this type are produced by the generalized Schur factorization of a complex matrix pair (A,B):

   A = Q*S*Z**H,  B = Q*P*Z**H

as computed by CGGHRD + CHGEQZ.

The right eigenvector x and the left eigenvector y of (S,P) corresponding to an eigenvalue w are defined by:

   S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

where y**H denotes the conjugate transpose of y. The eigenvalues are not input to this routine, but are computed directly from the diagonal elements of S and P. (Not me passing you nothing and expecting eigenvectors back 😌)

This routine returns the matrices X and/or Y of right and left eigenvectors of (S,P), or the products Z*X and/or Q*Y, where Z and Q are input matrices. If Q and Z are the unitary factors from the generalized Schur factorization of a matrix pair (A,B), then Z*X and Q*Y are the matrices of right and left eigenvectors of (A,B). Slay. 💅

📋 Parameters 📋

SIDE — CHARACTER*1, choose your fighter:
- = 'R': compute right eigenvectors only;
- = 'L': compute left eigenvectors only;
- = 'B': compute both right and left eigenvectors.
HOWMNY — CHARACTER*1, the vibe selector:
- = 'A': compute all right and/or left eigenvectors;
- = 'B': compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL;
- = 'S': compute selected right and/or left eigenvectors, specified by the logical array SELECT.
SELECT — LOGICAL array, dimension (N). If HOWMNY='S', SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the j-th eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A' or 'B'.
N — INTEGER. The order of the matrices S and P. N >= 0.
S — COMPLEX array, dimension (LDS,N). The upper triangular matrix S from a generalized Schur factorization, as computed by CHGEQZ.
LDS — INTEGER. The leading dimension of array S. LDS >= max(1,N).
P — COMPLEX array, dimension (LDP,N). The upper triangular matrix P from a generalized Schur factorization, as computed by CHGEQZ. P must have real diagonal elements.
LDP — INTEGER. The leading dimension of array P. LDP >= max(1,N).
VL — COMPLEX array, dimension (LDVL,MM).
- On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the unitary matrix Q of left Schur vectors returned by CHGEQZ).
- On exit, if SIDE = 'L' or 'B', VL contains:
  - if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
  - if HOWMNY = 'B', the matrix Q*Y;
  - if HOWMNY = 'S', the left eigenvectors of (S,P) specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues.
- Not referenced if SIDE = 'R'.
LDVL — INTEGER. The leading dimension of array VL. LDVL >= 1, and if SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
VR — COMPLEX array, dimension (LDVR,MM).
- On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (usually the unitary matrix Z of right Schur vectors returned by CHGEQZ).
- On exit, if SIDE = 'R' or 'B', VR contains:
  - if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
  - if HOWMNY = 'B', the matrix Z*X;
  - if HOWMNY = 'S', the right eigenvectors of (S,P) specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues.
- Not referenced if SIDE = 'L'.
LDVR — INTEGER. The leading dimension of array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
MM — INTEGER. The number of columns in the arrays VL and/or VR. MM >= M.
M — INTEGER. The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column.
WORK — COMPLEX array, dimension (2*N).
RWORK — REAL array, dimension (2*N).
INFO

okay so INFO is INTEGER:

= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌸 subroutine ctgexc 🌸

   subroutine ctgexc (logical WANTQ, logical WANTZ, integer N, complex,
       dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B,
       integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex,
       dimension( ldz, * ) Z, integer LDZ, integer IFST, integer ILST, integer
       INFO)

CTGEXC

✨ Purpose ✨

no cap, CTGEXC reorders the generalized Schur decomposition of a complex matrix pair (A,B), using an unitary equivalence transformation (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with row index IFST is moved to row ILST. it's giving glow-up for your matrix pair 💅

(A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. lowkey important.

Optionally, the matrices Q and Z of generalized Schur vectors are updated. obsessed with optional updates.

                   Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
                   Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

🎀 Parameters 🎀

WANTQ — WANTQ is LOGICAL
- .TRUE. : update the left transformation matrix Q;
- .FALSE.: do not update Q.
WANTZ — WANTZ is LOGICAL
- .TRUE. : update the right transformation matrix Z;
- .FALSE.: do not update Z.
N — N is INTEGER
- The order of the matrices A and B. N >= 0.
A — A is COMPLEX array, dimension (LDA,N)
- On entry, the upper triangular matrix A in the pair (A, B).
- On exit, the updated matrix A.
LDA — LDA is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX array, dimension (LDB,N)
- On entry, the upper triangular matrix B in the pair (A, B).
- On exit, the updated matrix B.
LDB — LDB is INTEGER
- The leading dimension of the array B. LDB >= max(1,N).
Q — Q is COMPLEX array, dimension (LDQ,N)
- On entry, if WANTQ = .TRUE., the unitary matrix Q.
- On exit, the updated matrix Q.
- If WANTQ = .FALSE., Q is not referenced.
LDQ — LDQ is INTEGER
- The leading dimension of the array Q. LDQ >= 1;
- If WANTQ = .TRUE., LDQ >= N.
Z — Z is COMPLEX array, dimension (LDZ,N)
- On entry, if WANTZ = .TRUE., the unitary matrix Z.
- On exit, the updated matrix Z.
- If WANTZ = .FALSE., Z is not referenced.
LDZ — LDZ is INTEGER
- The leading dimension of the array Z. LDZ >= 1;
- If WANTZ = .TRUE., LDZ >= N.
IFST — IFST is INTEGER
ILST — ILST is INTEGER
- Specify the reordering of the diagonal blocks of (A, B). The block with row index IFST is moved to row ILST, by a sequence of swapping between adjacent blocks.
INFO — INFO is INTEGER
- =0: Successful exit.
- <0: if INFO = -i, the i-th argument had an illegal value.
- =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill-conditioned. (A, B) may have been partially reordered, and ILST points to the first row of the current position of the block being moved.

careful, bestie: if INFO = 1, the problem is ill-conditioned and (A, B) may have been partially reordered. not a vibe 😭

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💖 Contributors: 💖

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. icons 💅

📚 References: 📚

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Author

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole thing, bestie 💖 This was Version 3.11.0, dated Sun Nov 27 2022, from the complexGEcomputational(3) manual page. periodt ✨💅

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complexGEcomputational(3)

💅 complexGEcomputational(3) — LAPACK 💅

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions

💖 Detailed Description 💖

🫶 Function Documentation 🫶

💅 subroutine cgebak

✨ Purpose ✨

🌸 Parameters 🌸

👑 Author 👑

💅 subroutine cgebal

✨ Purpose ✨

🌸 Parameters 🌸

🎀 Author 🎀

✨ Further Details ✨

🦋 subroutine cgebd2 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( * ) WORK, integer INFO)

Purpose ✨

Parameters 🎀

Author 💜

Further Details 💫

🌈 subroutine cgebrd (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( * ) WORK, integer LWORK, integer INFO)

Purpose ✨

Parameters 🎀

Author 👑

Further Details:

✨ CGECON ✨

🌸 CGEEQU 🌸

🦋 CGEEQUB 🦋

💜 CGEHD2 💜

A 🦋

LDA

TAU

WORK

INFO

👑 Author

✨ Further Details ✨

💖 subroutine cgehrd 💖

🎯 Purpose:

N

ILO

IHI

A

LDA

TAU

WORK

LWORK

INFO

👑 Author

🔍 Further Details:

💜 subroutine cgelq2 💜

🎯 Purpose:

M

N

A

LDA

TAU

WORK

INFO

👑 Author

🔍 Further Details:

🎀 subroutine cgelqf 🎀

🎯 Purpose:

M

N

A

LDA

TAU

WORK

LWORK

👑 Author

🌸 Further Details

🦋 subroutine cgemqrt

💖 Purpose

🎀 Parameters

🌈 Author

🧚 subroutine cgeql2

⭐ Purpose

🌸 Parameters

👑 Author

`A` 🦋

`LDA`

`TAU`

`WORK`

`INFO`

💖 `subroutine cgehrd` 💖

`N`

`ILO`

`IHI`

`A`

`LDA`

`TAU`

`WORK`

`LWORK`

`INFO`

💜 `subroutine cgelq2` 💜

`M`

`N`

`A`

`LDA`

`TAU`

`WORK`

`INFO`

🎀 `subroutine cgelqf` 🎀

`M`

`N`

`A`

`LDA`

`TAU`

`WORK`

`LWORK`

🌈 subroutine `cgeqr2` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

🦋 subroutine `cgeqr2p` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

🌈 subroutine `cgeqrf` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

🦋 subroutine `cgeqrfp` (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

🎀 `ERR_BNDS_COMP` 🎀

✨ `NPARAMS` ✨

✨ `PARAMS` ✨

✨ `WORK` ✨

✨ `RWORK` ✨

✨ `INFO` ✨

🌈 `subroutine cgerq2` 🌈

💜 `M`

💜 `N`

💜 `A`

💜 `LDA`

💜 `TAU`