realGEcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

, in debian bookworm

🌸 NAME 🌸

realGEcomputational — real 💖

✨ SYNOPSIS ✨

okay so here's the whole squad of functions, no cap:

subroutine sgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO) — SGEBAK
subroutine sgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO) — SGEBAL
subroutine sgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) — SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine sgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) — SGEBRD
subroutine sgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO) — SGECON
subroutine sgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) — SGEEQU
subroutine sgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) — SGEEQUB
subroutine sgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO) — SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine sgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO) — SGEHRD
subroutine sgelq2 (M, N, A, LDA, TAU, WORK, INFO) — SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine sgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — SGELQF
subroutine sgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO) — SGEMQRT
subroutine sgeql2 (M, N, A, LDA, TAU, WORK, INFO) — SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
subroutine sgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — SGEQLF
subroutine sgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO) — SGEQP3
subroutine sgeqr2 (M, N, A, LDA, TAU, WORK, INFO) — SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
subroutine sgeqr2p (M, N, A, LDA, TAU, WORK, INFO) — SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
subroutine sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — SGEQRF
subroutine sgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO) — SGEQRFP
subroutine sgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO) — SGEQRT
subroutine sgeqrt2 (M, N, A, LDA, T, LDT, INFO) — SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
recursive subroutine sgeqrt3 (M, N, A, LDA, T, LDT, INFO) — SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine sgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) — SGERFS
subroutine sgerfsx (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, IWORK, INFO) — SGERFSX
subroutine sgerq2 (M, N, A, LDA, TAU, WORK, INFO) — SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine sgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) — SGERQF
subroutine sgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO) — SGESVJ
subroutine sgetf2 (M, N, A, LDA, IPIV, INFO) — SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
subroutine sgetrf (M, N, A, LDA, IPIV, INFO) — SGETRF
recursive subroutine sgetrf2 (M, N, A, LDA, IPIV, INFO) — SGETRF2
subroutine sgetri (N, A, LDA, IPIV, WORK, LWORK, INFO) — SGETRI
subroutine sgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO) — SGETRS
subroutine shgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO) — SHGEQZ
subroutine sla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
real function sla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) — SLA_GERCOND estimates the Skeel condition number for a general matrix.
subroutine sla_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) — SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function sla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF) — SLA_GERPVGRW
subroutine slaorhr_col_getrfnp (M, N, A, LDA, D, INFO) — SLAORHR_COL_GETRFNP
recursive subroutine slaorhr_col_getrfnp2 (M, N, A, LDA, D, INFO) — SLAORHR_COL_GETRFNP2
subroutine stgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO) — STGEVC
subroutine stgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO) — STGEXC

💖 Detailed Description 💖

This is the group of real computational functions for GE matrices. periodt. 💅

🦋 Function Documentation 🦋

🎀 subroutine sgebak — SGEBAK 🎀

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

Purpose: bestie, SGEBAK forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by SGEBAL. That's literally the whole point. ✨

Parameters

JOB — JOB is CHARACTER*1. It 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 SGEBAL.
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 SGEBAL. 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 SGEBAL.
M — M is INTEGER. The number of columns of the matrix V. M >= 0.
V — V is REAL array, dimension (LDV,M). On entry, the matrix of right or left eigenvectors to be transformed, as returned by SHSEIN or STREVC. 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 sgebal — SGEBAL 🎀

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

Purpose: SGEBAL balances a general real matrix A. This involves, first, permuting 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; and second, applying 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 REAL 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 REAL array, dimension (N). Details of the permutations and scaling factors applied to A. If P(j) is the index of the row and column interchanged with row and column j and D(j) is the scaling factor applied to row and column j, then

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

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

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 permutations consist of row and column interchanges which put the matrix in the form

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

...where T1 and T2 are upper triangular matrices and their eigenvalues are just sitting there, right along the diagonal. 💅

The column indices ILO and IHI mark the starting and ending columns of the submatrix B, okay? And balancing? Bestie, balancing is literally just applying a diagonal similarity transformation inv(D) * B * D so that the 1-norms of each row of B and its corresponding column are nearly equal. That's the whole assignment. ✨

The output matrix comes out looking like this:

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

And all the info about the permutations P and the diagonal matrix D? It gets handed back to you in the vector SCALE. 👑

This subroutine is based on the EISPACK routine BALANC.

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

🦋 `sgebd2` 🦋

subroutine sgebd2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) D, real, dimension( * ) E, real,
    dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * )
    WORK, integer INFO)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. (the unblocked one, iykyk — it's giving small-and-simple energy 🌸)

✨ Purpose ✨

SGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

🎀 Parameters 🎀

M — M is INTEGER. The number of rows in the matrix A. M >= 0.
N — N is INTEGER. The number of columns in the matrix A. N >= 0.
A — A is REAL 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.

See Further Details. 👀

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
D — D is REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E — E is 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 — TAUQ is REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
TAUP — TAUP is REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
WORK — WORK is REAL array, dimension (max(M,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 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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 — look at the glow-up fr fr:

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). 🦋

🌈 `sgebrd` 🌈

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

SGEBRD — the blocked glow-up of the same job. 👑

✨ Purpose ✨

SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

🎀 Parameters 🎀

M — M is INTEGER. The number of rows in the matrix A. M >= 0.
N — N is INTEGER. The number of columns in the matrix A. N >= 0.
A — A is REAL 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.

See Further Details. 👀

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
D — D is REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E — E is 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 — TAUQ is REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
TAUP — TAUP is REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
WORK — WORK is REAL array, dimension (MAX(1,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,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. 🕵️‍♀️

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 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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). bestie that's the whole shape of it 💅

🌸 `sgecon` — SGECON 🌸

subroutine sgecon (character NORM, integer N, real, dimension( lda, * ) A,
    integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer,
    dimension( * ) IWORK, integer INFO)

🎯 Purpose

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

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 REAL array, dimension (LDA,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
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 REAL array, dimension (4*N).
IWORK — IWORK is INTEGER 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.

🦋 `sgeequ` — SGEEQU 🦋

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

🎯 Purpose

SGEEQU 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. fr fr, it's giving "best effort" energy 😌

📋 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 REAL 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.

✨ `sgeequb` — SGEEQUB ✨

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

🎯 Purpose

SGEEQUB 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 SGEEQU 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). the way it trades tidy magnitudes for zero extra rounding?? obsessed 😭

📋 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 REAL 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.

🎀 `sgehd2` — SGEHD2 🎀

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

subroutine sgehd2 (integer N, integer ILO, integer IHI, real, dimension(
    lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * )
    WORK, integer INFO)

🎯 Purpose

SGEHD2 reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation:

   Q**T * 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 SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= max(1,N).
A — A is REAL 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 orthogonal 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 REAL array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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 (ihi-ilo) elementary reflectors

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

okay so each H(i) has the form

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

where tau is a real scalar, and v is a real 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). ✨

🌸 `subroutine sgehrd` 🌸

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

SGEHRD 👑

Purpose:

SGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q**T * A * Q = H. no cap, that's literally the whole job 💖

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 SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A — A is REAL 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 orthogonal 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 REAL 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 REAL 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. 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**T

where tau is a real scalar, and v is a real 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 SGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See SLAHR2.) 📌

🎀 `subroutine sgelq2` 🎀

subroutine sgelq2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer INFO)

SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm. bestie this one is the unblocked version, okay, we're doing it the simple way today 💖

Purpose:

SGELQ2 computes an LQ factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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(2) H(1), where k = min(m,n).

Each H(i) has the form

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

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

🌈 `subroutine sgelqf` 🌈

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

SGELQF 🧚

Purpose:

SGELQF computes an LQ factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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. 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**T

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

💫 `subroutine sgemqrt` 💫

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

SGEMQRT 👑

Purpose:

SGEMQRT overwrites the general real M-by-N matrix C with

                SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q C            C Q
TRANS = 'T':   Q**T C            C Q**T

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

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

generated using the compact WY representation as returned by <code>SGEQRT</code>.

And <code>Q</code> is of order <code>M</code> if <code>SIDE = 'L'</code>, and of order <code>N</code> if <code>SIDE = 'R'</code>. okay bestie that's the shape lore 💅

Parameters ✨

SIDE — SIDE is CHARACTER*1
- = 'L': apply Q or Q**T from the Left;
- = 'R': apply Q or Q**T from the Right.
TRANS — TRANS is CHARACTER*1
- = 'N': No transpose, apply Q;
- = 'T': Transpose, apply Q**T.
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 SGEQRT. fr fr, don't switch it up, they have to match 💀
V — V is REAL 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 SGEQRT 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 REAL array, dimension (LDT,K) — the upper triangular factors of the block reflectors as returned by SGEQRT, stored as a NB-by-N matrix.
LDT — LDT is INTEGER — the leading dimension of the array T. LDT >= NB.
C — C is REAL array, dimension (LDC,N) — on entry, the M-by-N matrix C. On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.
LDC — LDC is INTEGER — the leading dimension of the array C. LDC >= max(1,M).
WORK — WORK is REAL 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 `sgeql2` 👑

subroutine sgeql2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer INFO)

<code>SGEQL2</code> computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose: 🦋

SGEQL2 computes a QL factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)) — the scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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**T

where tau is a real scalar, and v is a real 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 `sgeqlf` ✨

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

<code>SGEQLF</code>

Purpose: 🦋

SGEQLF computes a QL factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)) — the scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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). 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. it's giving "just asking, no commitment" energy 🎀

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**T

where tau is a real scalar, and v is a real 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 `sgeqp3` 🔥

subroutine sgeqp3 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU,
    real, dimension( * ) WORK, integer LWORK, integer INFO)

<code>SGEQP3</code>

Purpose: 🦋

SGEQP3 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 REAL 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 orthogonal 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. (yes bestie, "the the" — we keep the lore exactly as it was written, no edits 🙈)
TAU — TAU is REAL array, dimension (min(M,N)) — the scalar factors of the elementary reflectors.
WORK — WORK is REAL 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 >= 3*N+1. For optimal performance LWORK >= 2*N+( N+1 )*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. same "no strings attached" vibe check 💅

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**T

where tau is a real 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 `sgeqr2` 💫

subroutine sgeqr2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer INFO)

okay bestie so SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm. no cap, it's the little one that does the work one reflector at a time. 🫶

Purpose

SGEQR2 computes a QR factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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**T

where tau is a real scalar, and v is a real 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 `sgeqr2p` 🌸

subroutine sgeqr2p (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer INFO)

and now for her sister SGEQR2P 💖 — she computes the QR factorization of a general rectangular matrix with non-negative diagonal elements, using an unblocked algorithm. same energy, but the diagonal is giving ✨all positive✨ vibes only.

Purpose

SGEQR2P computes a QR factorization of a real 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 REAL 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 nonnegative; the elements below the diagonal, with the array TAU, represent the orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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**T

where tau is a real scalar, and v is a real 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. 📓 (iykyk)

subroutine `sgeqrf` 🔥

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

SGEQRF 👑 — this is the blocked glow-up, the one that actually gets called in real life. it's giving main character energy.

Purpose

SGEQRF computes a QR factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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. 🕵️‍♀️ (sneaky little peek, she won't yell at 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**T

where tau is a real scalar, and v is a real 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 `sgeqrfp` 🦋

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

SGEQRFP 🌈 — blocked and non-negative diagonal. the full package, honestly.

Purpose

SGEQR2P computes a QR factorization of a real 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 REAL 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 nonnegative; the elements below the diagonal, with the array TAU, represent the orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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). 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(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

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

where tau is a real scalar, and v is a real 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 `sgeqrt` 🎀

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

SGEQRT ⭐ — she's the block-reflector one, and we're just getting started with her...

Purpose

✨ Parameters ✨

M — M is INTEGER, bestie. It's the number of rows in the matrix A, and it better be M >= 0. 💅
N — N is INTEGER. Number of columns of A. N >= 0, obviously. 🌸
NB — NB is INTEGER, and it's the block size used in the blocked QR. The vibe check here is MIN(M,N) >= NB >= 1. 🔥
A — A is a REAL array, dimension (LDA,N).
- On entry: the M-by-N matrix A.
- On exit: the elements on and above the diagonal of the array hold the min(M,N)-by-N upper trapezoidal matrix R (and R is straight-up upper triangular if M >= N); the elements below the diagonal are the columns of V. She's doing double duty, iconic. 👑
LDA — LDA is INTEGER, the leading dimension of the array A. Requirement: LDA >= max(1,M). Don't play with this one. 😭
T — T is a REAL array, dimension (LDT,MIN(M,N)). It's the upper triangular block reflectors, stored in compact form as a sequence of upper triangular blocks. Peep below for the deets. 🎀
LDT — LDT is INTEGER, the leading dimension of the array T. LDT >= NB. Periodt.
WORK — WORK is a REAL array, dimension (NB*N). Just workspace, she's doing her best. 🫶
INFO — INFO is INTEGER.
- = 0: successful exit, we ate. ✨
- < 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. Not me pointing that out twice, but it matters. 💀

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, an 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 sgeqrt2

subroutine sgeqrt2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. She's giving classic, no recursion needed. 💅

💖 Purpose:

SGEQRT2 computes a QR factorization of a real 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. Requirement: M >= N. (Note that this is a stricter vibe than plain SGEQRT had, bestie. 👀)
N — N is INTEGER, the number of columns of the matrix A. N >= 0.
A — A is a REAL array, dimension (LDA,N).
- On entry: the real 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 REAL array, dimension (LDT,N). It's 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 (they're just sitting there, unemployed 🙈). 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**T

where V**T is the transpose of V. Simple as that, and honestly? Slay. ✨

🧚 recursive subroutine sgeqrt3

recursive subroutine sgeqrt3 (integer M, integer N, real, dimension( lda, *
) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer
INFO)

SGEQRT3</purpose> — okay not a real tag, but: SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. She recurses. She's main character energy. 👑

💖 Purpose:

SGEQRT3 recursively computes a QR factorization of a real 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. Cite your sources, bestie. 📚

✨ 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 REAL array, dimension (LDA,N).
- On entry: the real 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 REAL array, dimension (LDT,N). The N-by-N upper triangular factor of the block reflector. Elements on and above the diagonal contain the block reflector T; 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 )

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

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above). Go read it, it's giving academia. 🎓

💜 subroutine sgerfs

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

SGERFS — the glow-up arc for your solution. ✨

💖 Purpose:

SGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. She's not just fixing your answer, she's telling you how wrong you were, lovingly. 🫶

✨ Parameters ✨

TRANS — TRANS is CHARACTER*1, and it 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 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 a REAL 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 a REAL array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
LDAF — LDAF is INTEGER, the leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is an INTEGER array, dimension (N). The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). Swapped. Shuffled. Gorgeous. 🔄
B — B is a REAL 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 a REAL array, dimension (LDX,NRHS).
- On entry: the solution matrix X, as computed by SGETRS.
- On exit: the improved solution matrix X. Glow-up complete. 💅
LDX — LDX is INTEGER, the leading dimension of the array X. LDX >= max(1,N).
FERR — FERR is a REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. So like, she's a little dramatic, but she's not wrong. 😭
BERR —
BERR — REAL array, dimension (NRHS). This is the componentwise relative backward error of every solution vector X(j). Translation bestie: it's the smallest relative change you could make to any element of A or B that would make X(j) an actual exact solution. tiny number = good vibes ✨
WORK — REAL array, dimension (3*N). Scratch space, don't touch it, it's busy 💅
IWORK — INTEGER array, dimension (N). Same deal, but integers.
INFO — INTEGER. The receipts:
- = 0: successful exit, we ate, no crumbs 👑
- < 0: if INFO = -i, then the i-th argument had an illegal value. so like. it's on you bestie 😭

Internal Parameters

ITMAX is the maximum number of steps of iterative refinement. (i.e. how many glow-up passes the algorithm is allowed to do on your solution before it gives up and hands it over as-is.)

Author

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

subroutine `sgerfsx`

subroutine sgerfsx (character TRANS, character EQUED, integer N, integer
    NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, *
    ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * )
    R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB,
    real, 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, real, dimension( * ) WORK, integer, dimension( *
    ) IWORK, integer INFO)

SGERFSX — yes the name is giving extra and it earned it. 💖

Purpose

SGERFSX improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. So it takes your solution, does a lil refinement arc, and also hands you receipts on how wrong it might be. Iconic behaviour, honestly. 🎀

On top of the normwise error bound, the code provides a maximum componentwise error bound if possible. Peep the comments on ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those bounds.

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED, R and C below. If that happened, the solution and error bounds returned are for the original unequilibrated system — like, it undoes the glow-up so you get your actual answer back. 🦋

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument entirely. Lazy? No. Efficient. periodt. 💅

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)
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 skimping, it matters fr fr.
- = '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 REAL array, dimension (LDA,N). The original N-by-N matrix A. the moment. the main character. 🌟
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is REAL array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
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 SGETRF; 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.

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. 🚨 bestie this one's a real caveat, don't skip it

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.

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. (same warning, different array, still very much a thing 😭)

B — B is REAL 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 REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X. she's been to the gym. 🏋️‍♀️
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 don't panic-spiral just because the number is smol 🙈

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: 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.

See Lapack Working Note 165 for further details and extra cautions. 📖

ERR_BNDS_COMP — ERR_BNDS_COMP 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 ERR_BNDS_COMP is not accessed. 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 — '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. This is the estimated forward error, and it's almost certainly within a factor of 10 of the true error, as long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). Bestie, only trust this one if the previous boolean is true. Like, that's the vibe check. 🙈
err = 3 — Reciprocal condition number: the estimated componentwise reciprocal condition number. It gets compared against the threshold sqrt(n) * slamch('Epsilon') to figure out whether 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. Iconic scaling behaviour, honestly. ✨

See Lapack Working Note 165 for further details and extra cautions. (Read it. It's giving "required reading." 👑)

NPARAMS

NPARAMS is INTEGER.

It specifies the number of parameters set in PARAMS. If it's <= 0, the PARAMS array is never referenced and default values are used. Like, don't even look at it. 💅

PARAMS

PARAMS is REAL array, dimension NPARAMS.

It 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. No cap. 🧚

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)
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. ⚠️ so like, proceed with caution, fr.
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 REAL array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

INFO is INTEGER

= 0: successful exit. The solution to every right-hand side is guaranteed. slay. 🎉
< 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 sgerq2` 🧚

subroutine sgerq2 (integer M, integer N, real, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK,
    integer INFO)

SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm. 💅

Purpose

SGERQ2 computes an RQ factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; 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). periodt. ✨

🦋 `subroutine sgerqf` 🦋

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

SGERQF

Purpose

SGERQF computes an RQ factorization of a real 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 REAL 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 orthogonal 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 REAL array, dimension (min(M,N)). The scalar factors of the elementary reflectors (see Further Details).
WORK — WORK is REAL 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 >= M, otherwise. For optimum performance LWORK >= M*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. (the sneak peek manoeuvre 💅)

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**T

where tau is a real scalar, and v is a real vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; 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 sgesvj` 💜

subroutine sgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV,
    integer M, integer N, real, dimension( lda, * ) A, integer LDA, real,
    dimension( n ) SVA, integer MV, real, dimension( ldv, * ) V, integer
    LDV, real, dimension( lwork ) WORK, integer LWORK, integer INFO)

SGESVJ

Purpose

Okay bestie, so SGESVJ computes the singular value decomposition (SVD) of a real M-by-N matrix A, where M >= N. 💖 The SVD of A is written like this:

                                    [++]   [xx]   [x0]   [xx]
              A = U * SIGMA * V^t,  [++] = [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 orthogonal 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. And here's the iconic part: SGESVJ can sometimes compute tiny singular values and their singular vectors much more accurately than other SVD routines — see below under Further Details. ✨

✨ Parameters ✨

🎀 JOBA

JOBA is CHARACTER*1.

It specifies the structure of A, bestie:

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

🎀 JOBU

JOBU is CHARACTER*1.

It says whether we're computing the left singular vectors (aka the columns of U):

= 'U': the left singular vectors corresponding to the nonzero singular values get computed and returned in the leading columns of A. More deets in the description of A. The default numerical orthogonality threshold is set to approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
= 'C': same vibe as JOBU='U', except you get to 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, periodt. CTOL greater than 1 / EPS is meaningless. The 'C' option 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.

It says whether we're computing the right singular vectors, that is, the matrix V:

= 'V': 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 REAL 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 WORK as RANKA=NINT(WORK(2)). Also see the descriptions of SVA and WORK. 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 SGESVJ 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^T||_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 SGESVJ 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 = WORK(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 SGESVJ 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 SGESVJ is applied to the first MV rows of V. See the description of JOBV.

🎀 V

V is REAL 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) .

🎀 WORK

WORK is REAL array, dimension (LWORK).

On entry:

If JOBU = 'C': WORK(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:

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

🎀 LWORK

LWORK is INTEGER.

Length of WORK, WORK >= MAX(6,M+N).

🎀 INFO

INFO is INTEGER.

= 0: successful exit. 🎉
< 0: if INFO = -i, then the i-th argument had an illegal value
> 0: SGESVJ did not converge in the maximal allowed number (30) of sweeps. The output may still be useful. See the description of WORK.

✨ Author ✨

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

🌸 Further Details 🌸

okay so the tea on the orthogonal N-by-N matrix V: it gets built as a product of Jacobi plane rotations, and those rotations are implemented as the fast scaled rotations of Anda and Park [1]. And if the Jacobi angle underflows? 💀 no worries, a modified Jacobi transformation of Drmac [4] slides in. The pivot strategy uses column interchanges of de Rijk [2]. The relative accuracy of the computed singular values, plus the accuracy of the computed singular vectors (in angle metric), is exactly as guaranteed by the theory of Demmel and Veselic [3]. The condition number that decides the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition number. Best performance of this Jacobi SVD procedure hits different when it's used in an accelerated version of Drmac and Veselic [5,6] — it's literally the kernel routine in the SIGMA library [7]. 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. 😭

🫶 Contributors 🫶

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

📚 References 📚

[1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.

SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.

[2] 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.

[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[4] 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.

[5] 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.

[6] 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.

[7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations.

Department of Mathematics, University of Zagreb, 2008.

🐛 Bugs, Examples and Comments 💌

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

👑 subroutine sgetf2 (`integer M`, `integer N`, `real, dimension( lda, * ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `integer INFO`) 👑

SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

🎯 Purpose

SGETF2 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

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 REAL 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

LDA is INTEGER

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

IPIV

IPIV is 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 is 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. 🚨

🖋️ Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

👑 subroutine sgetrf (`integer M`, `integer N`, `real, dimension( lda, * ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `integer INFO`) 👑

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

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

🎯 Purpose

SGETRF 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

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 REAL 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

LDA is INTEGER

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

IPIV

IPIV is 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 is 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 (round two, bestie)

SGETRF 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

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 REAL 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

LDA is INTEGER

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

IPIV

IPIV is 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 is 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

🎯 Purpose (round three, the iterative one)

SGETRF 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 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.

The pattern is as follows, with the large blocks of U being updated in one call to STRSM, 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. 💅

[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

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 REAL 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
LDA — INTEGER, bestie. The leading dimension of the array A. Rule of the land: LDA >= max(1,M). Do not test it. 🫶
IPIV — INTEGER array, dimension (min(M,N)). These are the pivot indices, and here's the tea: for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i). Every row swap, receipted. 🔄
INFO — INTEGER, the final vibe check:
- = 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

recursive subroutine `sgetrf2`

recursive subroutine sgetrf2 (integer M, integer N, real, dimension( lda, * )
A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

SGETRF2

Purpose 💫

SGETRF2 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

The subroutine calls itself to factor:

                                      [ A11 ]
The subroutine calls itself to factor [ --- ],
                                      [ A12 ]

and then it does the swaps on:

                [ A12 ]
do the swaps on [ --- ], solve A12, update A22,
                [ A22 ]

then calls itself to factor A22 and do the swaps on A21. Recursion, but make it math. 🌀

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 — REAL 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.

subroutine `sgetri`

subroutine sgetri (integer N, real, dimension( lda, * ) A, integer LDA,
integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK,
integer INFO)

SGETRI

Purpose 💫

SGETRI computes the inverse of a matrix using the LU factorization computed by SGETRF.

This method inverts U and then computes inv(A) by solving the system inv(A)*L = inv(U) for inv(A). It's giving "undo the factoring, get the original back." 🪄

Parameters

N — INTEGER. The order of the matrix A. N >= 0.
A — REAL array, dimension (LDA,N). On entry, the factors L and U from the factorization A = P*L*U as computed by SGETRF. On exit, if INFO = 0, the inverse of the original matrix A.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
WORK — REAL 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. No inverse for you, bestie. 💀

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine `sgetrs`

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

SGETRS

Purpose 💫

SGETRS solves a system of linear equations

   A * X = B  or  A**T * X = B

with a general N-by-N matrix A using the LU factorization computed by SGETRF. Feed it the factorization, get the solution. That's the whole gig. 🌸

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**T* X = B (Conjugate transpose = 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 — REAL array, dimension (LDA,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
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.

subroutine `shgeqz`

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

SHGEQZ

Purpose 💫

SHGEQZ computes the eigenvalues of a real 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 real matrix pair (A,B):

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

as computed by SGGHRD.

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

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

where Q and Z are orthogonal matrices, P is an upper triangular matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues. Two flavors of eigenvalue, one matrix pair. 😌

Additionally, the 2-by-2 upper triangular diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0, P(j,j) > 0, and P(j+1,j+1) > 0.

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

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

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.

Real eigenvalues can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Just read it off the diagonal, bestie. 🎀

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

Parameters

JOB — CHARACTER*1:
- = 'E': Compute eigenvalues only;
- = 'S': Compute eigenvalues and the Schur form.
COMPQ

COMPQ

COMPQ is CHARACTER*1, bestie, and here's the tea:

= 'N': Left Schur vectors (Q) are not computed. Not today, not ever.
= 'I': Q gets initialized to the unit matrix, and the matrix Q of left Schur vectors of (H,T) is handed back to you 💅
= 'V': Q must contain an orthogonal matrix Q1 on entry, and then you get the product Q1*Q returned. So like, you gotta bring your own Q1, okay?

COMPZ

COMPZ is also CHARACTER*1, and she's the mirror sister:

= '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 (H,T) is returned ✨
= 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.

N

N is INTEGER — the order of the matrices H, T, Q, and Z. And N >= 0, periodt.

ILO

ILO is INTEGER. That's it, that's the entry. (She gets explained in IHI's paragraph, iykyk.)

IHI

IHI is INTEGER.

So like, ILO and IHI mark the rows and columns of H which are in Hessenberg form. It's assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. And: if N > 0, then 1 <= ILO <= IHI <= N; but if N = 0, then ILO=1 and IHI=0. Yes queen, the empty case is real and it's handled 🫶

H

H is REAL array, dimension (LDH, N).

On entry, it's the N-by-N upper Hessenberg matrix H. On exit, if JOB = 'S', H contains the upper quasi-triangular matrix S from the generalized Schur factorization. But if JOB = 'E', the diagonal blocks of H match those of S, and the rest of H is just… unspecified 🙈 (do not look at her)

LDH

LDH is INTEGER — the leading dimension of the array H. And LDH >= max( 1, N ), no cap.

T

T is REAL array, dimension (LDT, N).

On entry, the N-by-N upper triangular matrix T. On exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization. And here's the fine print, so listen up: the 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if H(j+1,j) is non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and T(j+1,j+1) > 0. Slay.

If JOB = 'E', the diagonal blocks of T match those of P, but the rest of T is unspecified. Again — eyes off the leftovers 💀

LDT

LDT is INTEGER — the leading dimension of the array T. LDT >= max( 1, N ).

ALPHAR

ALPHAR is REAL array, dimension (N) — the real parts of each scalar alpha defining an eigenvalue of GNEP ✨

ALPHAI

ALPHAI is REAL array, dimension (N) — the imaginary parts of each scalar alpha defining an eigenvalue of GNEP.

And the drama of it: if ALPHAI(j) is zero, then the j-th eigenvalue is real. If it's positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). They come as a matching set, bestie, never separate them 🦋

BETA

BETA is REAL array, dimension (N) — the scalars beta that define the eigenvalues of GNEP.

Together, the quantities alpha = (ALPHAR(j),ALPHAI(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. And the warning, okay, highlight it: since either lambda or mu may overflow, they should not, in general, be computed. Do not do the division. I am begging 🙏

Q

Q is REAL array, dimension (LDQ, N).

On entry, if COMPQ = 'V', it's the orthogonal matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', it's the orthogonal matrix of left Schur vectors of (H,T), and if COMPQ = 'V', it's the orthogonal matrix of left Schur vectors of (A,B).

Not referenced if COMPQ = 'N'. She's off duty, leave her alone 💅

LDQ

LDQ is INTEGER — the leading dimension of the array Q. LDQ >= 1. But if COMPQ='V' or 'I', then LDQ >= N. Conditional requirements, love that for us.

Z

Z is REAL array, dimension (LDZ, N).

On entry, if COMPZ = 'V', it's the orthogonal matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', it's the orthogonal matrix of right Schur vectors of (H,T), and if COMPZ = 'V', it's the orthogonal 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. Same energy as LDQ, they're a duo 🎀

WORK

WORK is REAL array, dimension (MAX(1,LWORK)).

On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. She's telling you what she wants, respect it 💖

LWORK

LWORK is INTEGER — the dimension of the array WORK. And LWORK >= max(1,N).

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. Sneaky little shortcut, iykyk 😌

INFO

INFO is INTEGER:

= 0: successful exit. We love a clean win 👑
< 0: if INFO = -i, the i-th argument had an illegal value. Rude of you.
= 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct. So like, partial credit 😭
= N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(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:

Iteration counters, bestie, keep up:

JITER — counts iterations.
IITER — counts iterations run since ILAST was last changed. This is therefore reset only when a 1-by-1 or 2-by-2 block deflates off the bottom. Only then. Not before. 🦋

subroutine sla_geamv

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

SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds. Like, she's not here for the math math, she's here for the error bounds ✨

Purpose:

SLA_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.

This function is primarily used in calculating error bounds. And now the protection detail: 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. She knows the difference between a real zero and a fake zero, no cap 💅

Parameters

TRANS

TRANS is 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. She stays exactly how you left her 🫶

M

M is INTEGER.

On entry, M specifies the number of rows of the matrix A. M must be at least zero. Unchanged on exit.

N

N is INTEGER.

On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.

ALPHA

ALPHA is REAL.

On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

A

A is REAL 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

LDA is 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

X is REAL 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 is INTEGER.

On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.

BETA

BETA is REAL.

On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit. (Free pass when it's zero, we love an efficient queen 🎀)

Y

Y is REAL array, dimension at least

( 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's changed, she's grown, she's the updated y now 💅

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. (say it with me: level twooooo)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

real function sla_gercond

real function sla_gercond (character TRANS, integer N, real, dimension( lda,
    * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF,
    integer, dimension( * ) IPIV, integer CMODE, real, dimension( * ) C,
    integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)

SLA_GERCOND estimates the Skeel condition number for a general matrix. Yes bestie, the Skeel one, she's fancy 💅

Purpose:

okay so, continuing from where we were 💖 — SLA_GERCOND estimates the Skeel condition number of op(A) * op2(C), and op2 is whatever CMODE says it is, bestie:

CMODE = 1 → op2(C) = C
CMODE = 0 → op2(C) = I
CMODE = -1 → op2(C) = inv(C)

The Skeel condition number cond(A) = norminf( |inv(A)||A| ) gets computed by finding scaling factors R such that diag(R)*A*op2(C) is row equilibrated, and then just computing the standard infinity-norm condition number. no cap, that's the whole move ✨

💖 Parameters 💖

TRANS — CHARACTER*1. Tells us 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 — INTEGER. The number of linear equations, aka the order of the matrix A. Gotta have N >= 0.
A — REAL array, dimension (LDA,N). On entry, the N-by-N matrix A. the moment ✨
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — REAL array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by SGETRF; row i of the matrix was interchanged with row IPIV(i). iconic behavior 🎀
CMODE — INTEGER. Determines op2(C) in the formula op(A) * op2(C) as follows:
- CMODE = 1 op2(C) = C
- CMODE = 0 op2(C) = I
- CMODE = -1 op2(C) = inv(C)
C — REAL array, dimension (N). The vector C in the formula op(A) * op2(C).
INFO — INTEGER.
- = 0: Successful exit. slay 🫶
- i > 0: The ith argument is invalid.
WORK — REAL array, dimension (3*N). Workspace. (gotta have somewhere to put the thoughts 💭)
IWORK — INTEGER array, dimension (N). Workspace. same vibe, but integers.

👑 Author 👑

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

🦋 `sla_gerfsx_extended` 🦋

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

SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. it's giving overachiever and we love that 💅

💜 Purpose 💜

SLA_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 SGERFSX to perform iterative refinement.

In addition to a normwise error bound, the code provides a maximum componentwise error bound if possible. See the comments for ERRS_N and ERRS_C for the deets on the error bounds. Note that this subroutine is only responsible for setting the second fields of ERRS_N and ERRS_C. (division of labor, bestie, we respect it 🫡)

✨ Parameters ✨

PREC_TYPE — 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
- P = 'D': Double
- P = 'I': Indigenous
- P = 'X' or 'E': Extra
TRANS_TYPE — 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 = 'T': Transpose
- T = 'C': Conjugate transpose
N — INTEGER. The number of linear equations, i.e., 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.
A — REAL array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — REAL array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.
LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by SGETRF; row i of the matrix was interchanged with row IPIV(i).
COLEQU — 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. (skip this and the math gets feral, fr fr)
C — 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.

⚠️ okay so that last bit is the important part: powers of the radix = safe, anything else = rounding errors, and then you're refining against a matrix that isn't even your matrix anymore and your error estimates are lowkey untrustworthy. don't do it 😭

B — REAL array, dimension (LDB,NRHS). The right-hand-side matrix B.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Y — REAL array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix Y. glow-up arc ✨
LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT — 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 SLA_LIN_BERR.

N_NORMS — 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 — 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. iykyk 📖

ERRS_C — 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.

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 🛡️. This is the estimated forward error, and it's 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'). Bestie listen: this error bound should only be trusted if the previous boolean is true. No cap.
err = 3 — the Reciprocal condition number 🌀. Estimated componentwise reciprocal condition number. It gets compared against the threshold sqrt(n) * slamch('Epsilon') to figure out whether 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, fr fr. That's its whole job.

See Lapack Working Note 165 for further details and extra cautions 👀.

🎀 RES

RES is REAL array, dimension (N) Workspace to hold the intermediate residual.

🌸 AYB

AYB is REAL array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL. Reuse is slay.

🦋 DY

DY is REAL array, dimension (N) Workspace to hold the intermediate solution.

⭐ Y_TAIL

Y_TAIL is REAL 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. Not me warning you twice but here we are.

🔥 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 💀.

🌈 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..

👑 INFO

INFO is INTEGER
- = 0: Successful exit. 🎉
- < 0: if INFO = -i, the ith argument to SGETRS had an illegal value.

💜 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ real function `sla_gerpvgrw` ✨

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

SLA_GERPVGRW 🫶

💖 Purpose:

SLA_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. Big yikes, watch out for that one.

🌟 Parameters

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

✨ subroutine `slaorhr_col_getrfnp` ✨

subroutine slaorhr_col_getrfnp (integer M, integer N, real, dimension( lda,
    * ) A, integer LDA, real, dimension( * ) D, integer INFO)

SLAORHR_COL_GETRFNP 🎀

💖 Purpose:

SLAORHR_COL_GETRFNP computes the modified LU factorization without pivoting of a real 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);
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 SORHR_COL. In SORHR_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 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 SLAORHR_COL_GETRFNP2. Recursion go brrr 🌀

[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 — INTEGER The number of rows of the matrix A. M >= 0.
N — INTEGER The number of columns of the matrix A. N >= 0.
A — REAL 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 — REAL 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 plus or minus one.
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

✨ recursive subroutine `slaorhr_col_getrfnp2` ✨

recursive subroutine slaorhr_col_getrfnp2 (integer M, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer
    INFO)

SLAORHR_COL_GETRFNP2 🦋

💖 Purpose:

SLAORHR_COL_GETRFNP2 computes the modified LU factorization without pivoting of a real 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);
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).

okay so this routine? it's literally the auxiliary bestie for the Householder reconstruction routine SORHR_COL 💖 In SORHR_COL, this routine gets applied to an M-by-N matrix A with orthonormal columns, where every element is bounded by one in absolute value. And with the choice of the matrix S above, you 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 no pivoting is required for numerical stability [1]. no cap.

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.

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

SLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked routine SLAORHR_COL_GETRFNP, which uses blocked code calling Level 3 BLAS to update the submatrix. However, SLAORHR_COL_GETRFNP2 is self-sufficient and can be used without SLAORHR_COL_GETRFNP. 💅

[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 — M is INTEGER. The number of rows of the matrix A. M >= 0. row count bestie 💖
N — N is INTEGER. The number of columns of the matrix A. N >= 0. column count slay ✨
A — A is REAL 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. the glow up is real 🦋
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). leading dimension vibes 👑
D — D is REAL 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 plus or minus one. only ±1, no in-between 💀
INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value. bestie if it's negative, something's wrong 😭

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

STGEVC ✨

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

Purpose: 💅

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

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

as computed by SGGHRD + SHGEQZ.

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 tranpose of y. The eigenvalues are not input to this routine, but are computed directly from the diagonal blocks of S and P. eigen everything 🌈

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 orthogonal 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). iconic behavior 👑

Parameters 🎀

SIDE — SIDE is CHARACTER*1. = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. pick your side bestie 💖
HOWMNY — HOWMNY is CHARACTER*1. = '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. how many? yes ✨
SELECT — SELECT is LOGICAL array, dimension (N). If HOWMNY='S', SELECT specifies the eigenvectors to be computed. If w(j) is a real eigenvalue, the corresponding real eigenvector is computed if SELECT(j) is .TRUE.. If w(j) and w(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to .FALSE.. Not referenced if HOWMNY = 'A' or 'B'. the chosen ones 🦋
N — N is INTEGER. The order of the matrices S and P. N >= 0. order up 👑
S — S is REAL array, dimension (LDS,N). The upper quasi-triangular matrix S from a generalized Schur factorization, as computed by SHGEQZ. quasi-triangular realness 💅
LDS — LDS is INTEGER. The leading dimension of array S. LDS >= max(1,N). leading dim, no skimping ⭐
P — P is REAL array, dimension (LDP,N). The upper triangular matrix P from a generalized Schur factorization, as computed by SHGEQZ. 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S must be in positive diagonal form. positive diagonal form only, periodt 💖
LDP — LDP is INTEGER. The leading dimension of array P. LDP >= max(1,N). another leading dim, another slay ✨
VL — VL is REAL 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 orthogonal matrix Q of left Schur vectors returned by SHGEQZ). 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. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. Not referenced if SIDE = 'R'. left side story 💜
LDVL — LDVL is INTEGER. The leading dimension of array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N. dimension check 🎀
VR — VR is REAL array, dimension (LDVR,MM). On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Z (usually the orthogonal matrix Z of right Schur vectors returned by SHGEQZ). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P); if HOWMNY = 'B' or 'b', the matrix Z*X; if HOWMNY = 'S' or 's', the right eigenvectors of (S,P) specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. Not referenced if SIDE = 'L'. right side story 💖
LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N. leading dim again, we love consistency 👑
MM — MM is INTEGER. The number of columns in the arrays VL and/or VR. MM >= M. max columns, got it ✨
M — M is 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 real eigenvector occupies one column and each selected complex eigenvector occupies two columns. columns used, no cap 💅
WORK — WORK is REAL array, dimension (6*N). workspace bestie 🦋
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex eigenvalue. if positive, that block is not giving complex eigenvalue energy 😭

Author 🌸

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

Further Details 🔥

Allocation of workspace:
---------- -- ---------

   WORK( j ) = 1-norm of j-th column of A, above the diagonal
   WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
   WORK( 2*N+1:3*N ) = real part of eigenvector
   WORK( 3*N+1:4*N ) = imaginary part of eigenvector
   WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
   WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

Rowwise vs. columnwise solution methods:
------- --  ---------- -------- -------

Finding a generalized eigenvector consists basically of solving the
singular triangular system

 (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

and that's the math, bestie 💖✨

okay so bestie, imagine we're finding the i-th right eigenvector (assume all eigenvalues are real). The equation to be solved is literally this:

                  n                   i
             0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
                 k=j                 k=j

where C = (A - w B) and the components v(i+1:n) are 0.

The 'rowwise' method is:

(1)  v(i) := 1
for j = i-1,. . .,1:
                        i
    (2) compute  s = - sum C(j,k) v(k)   and
                      k=j+1

    (3) v(j) := s / C(j,j)

Step 2 is sometimes called the 'dot product' step, since it is an inner product between the j-th row and the portion of the eigenvector that has been computed so far.

The 'columnwise' method consists basically in doing the sums for all the rows in parallel. As each v(j) is computed, the contribution of v(j) times the j-th column of C is added to the partial sums. Since FORTRAN arrays are stored columnwise, this has the advantage that at each step, the elements of C that are accessed are adjacent to one another, whereas with the rowwise method, the elements accessed at a step are spaced LDS (and LDP) words apart.

When finding left eigenvectors, the matrix in question is the transpose of the one in storage, so the rowwise method then actually accesses columns of A and B at each step, and so is the preferred method.

subroutine stgexc

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

STGEXC

Purpose:

STGEXC reorders the generalized real Schur decomposition of a real matrix pair (A,B) using an orthogonal equivalence transformation

               (A, B) = Q * (A, B) * Z**T,

so that the diagonal block of (A, B) with row index IFST is moved to row ILST.

(A, B) must be in generalized real Schur canonical form (as returned by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are updated.

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

Parameters

WANTQ — WANTQ is LOGICAL, bestie. .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. It's the order of the matrices A and B. N >= 0, no cap.
A — A is REAL array, dimension (LDA,N). On entry, the matrix A in generalized real Schur canonical form. On exit, the updated matrix A, again in generalized real Schur canonical form. iconic.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is REAL array, dimension (LDB,N). On entry, the matrix B in generalized real Schur canonical form (A,B). On exit, the updated matrix B, again in generalized real Schur canonical form (A,B).
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Q — Q is REAL array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., the orthogonal 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 REAL array, dimension (LDZ,N). On entry, if WANTZ = .TRUE., the orthogonal 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. It specifies 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. On exit, if IFST pointed on entry to the second row of a 2-by-2 block, it is changed to point to the first row; ILST always points to the first row of the block in its final position (which may differ from its input value by +1 or -1). 1 <= IFST, ILST <= N.
WORK — WORK is REAL 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 when N <= 1, otherwise LWORK >= 4*N + 16. 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. =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.

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.

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.

Author Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole vibe, bestie! this page is from LAPACK version 3.11.0, dated Sun Nov 27 2022, for realGEcomputational(3). 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

realGEcomputational(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

💖 Detailed Description 💖

🦋 Function Documentation 🦋

🎀 subroutine sgebak — SGEBAK 🎀

Parameters

Author

🎀 subroutine sgebal — SGEBAL 🎀

Parameters

Author

Further Details

🦋 sgebd2 🦋

✨ Purpose ✨

🎀 Parameters 🎀

🫶 Author 🫶

💅 Further Details 💅

🌈 sgebrd 🌈

✨ Purpose ✨

🎀 Parameters 🎀

🫶 Author 🫶

💅 Further Details 💅

🌸 sgecon — SGECON 🌸

🎯 Purpose

📋 Parameters

👑 Author

🦋 sgeequ — SGEEQU 🦋

🎯 Purpose

📋 Parameters

👑 Author

✨ sgeequb — SGEEQUB ✨

🎯 Purpose

📋 Parameters

👑 Author

🎀 sgehd2 — SGEHD2 🎀

🎯 Purpose

📋 Parameters

👑 Author

🔍 Further Details

🌸 subroutine sgehrd 🌸

🎀 subroutine sgelq2 🎀

🌈 subroutine sgelqf 🌈

💫 subroutine sgemqrt 💫

Parameters ✨

Author 🌸

subroutine sgeql2 👑

Parameters

Author 🌸

Further Details: 💜

subroutine sgeqlf ✨

Parameters

Author 🌸

Further Details: 💜

subroutine sgeqp3 🔥

Parameters

Author 🌸

Further Details: 💜

Contributors: 🌈

subroutine sgeqr2 💫

Purpose

Parameters

Author

Further Details

subroutine sgeqr2p 🌸

Purpose

Parameters

Author

Further Details

subroutine sgeqrf 🔥

Purpose

Parameters

Author

Further Details

subroutine sgeqrfp 🦋

Purpose

Parameters

Author

Further Details

subroutine sgeqrt 🎀

Purpose

🦋 `sgebd2` 🦋

🌈 `sgebrd` 🌈

🌸 `sgecon` — SGECON 🌸

🦋 `sgeequ` — SGEEQU 🦋

✨ `sgeequb` — SGEEQUB ✨

🎀 `sgehd2` — SGEHD2 🎀

🌸 `subroutine sgehrd` 🌸

🎀 `subroutine sgelq2` 🎀

🌈 `subroutine sgelqf` 🌈

💫 `subroutine sgemqrt` 💫

subroutine `sgeql2` 👑

subroutine `sgeqlf` ✨

subroutine `sgeqp3` 🔥

subroutine `sgeqr2` 💫

subroutine `sgeqr2p` 🌸

subroutine `sgeqrf` 🔥

subroutine `sgeqrfp` 🦋

subroutine `sgeqrt` 🎀

subroutine `sgerfsx`

🧚 `subroutine sgerq2` 🧚

🦋 `subroutine sgerqf` 🦋

💜 `subroutine sgesvj` 💜

👑 subroutine sgetf2 (`integer M`, `integer N`, `real, dimension( lda, * ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `integer INFO`) 👑

👑 subroutine sgetrf (`integer M`, `integer N`, `real, dimension( lda, * ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `integer INFO`) 👑