complex16GBcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

, in debian bookworm

💖 `complex16GBcomputational` (3) — LAPACK 🎀

okay so bestie, you've wandered into the band-matrix corner of LAPACK and honestly? it's a whole vibe 💅

✨ NAME ✨

complex16GBcomputational — complex16

🌸 SYNOPSIS 🌸

Functions

zgbbrd — ZGBBRD

subroutine zgbbrd (VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, LDQ, PT,
    LDPT, C, LDC, WORK, RWORK, INFO)

zgbcon — ZGBCON

subroutine zgbcon (NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK,
    RWORK, INFO)

zgbequ — ZGBEQU

subroutine zgbequ (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX,
    INFO)

zgbequb — ZGBEQUB

subroutine zgbequb (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX,
    INFO)

zgbrfs — ZGBRFS

subroutine zgbrfs (TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
    B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)

zgbrfsx — ZGBRFSX

subroutine zgbrfsx (TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
    IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
    ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)

zgbtf2 — ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. no blocking, just raw unblocked energy ✨

subroutine zgbtf2 (M, N, KL, KU, AB, LDAB, IPIV, INFO)

zgbtrf — ZGBTRF

subroutine zgbtrf (M, N, KL, KU, AB, LDAB, IPIV, INFO)

zgbtrs — ZGBTRS

subroutine zgbtrs (TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)

zggbak — ZGGBAK

subroutine zggbak (JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV,
    INFO)

zggbal — ZGGBAL

subroutine zggbal (JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
    WORK, INFO)

zla_gbamv — ZLA_GBAMV performs a matrix-vector operation to calculate error bounds. it's giving "i'm here to do math for error bounds" 💖

subroutine zla_gbamv (TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX,
    BETA, Y, INCY)

zla_gbrcond_c — ZLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.

double precision function zla_gbrcond_c (TRANS, N, KL, KU, AB, LDAB,
    AFB, LDAFB, IPIV, C, CAPPLY, INFO, WORK, RWORK)

zla_gbrcond_x — ZLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.

double precision function zla_gbrcond_x (TRANS, N, KL, KU, AB, LDAB,
    AFB, LDAFB, IPIV, X, INFO, WORK, RWORK)

zla_gbrfsx_extended — ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. extra precise. extra refined. extra everything 👑

subroutine zla_gbrfsx_extended (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS,
    AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT,
    N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND,
    ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)

zla_gbrpvgrw — ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.

double precision function zla_gbrpvgrw (N, KL, KU, NCOLS, AB, LDAB, AFB,
    LDAFB)

zungbr — ZUNGBR

subroutine zungbr (VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO)

🦋 DETAILED DESCRIPTION 🦋

This is the group of complex16 computational functions for GB matrices ✨ (GB = general band, iykyk)

👑 FUNCTION DOCUMENTATION 👑

💅 subroutine `zgbbrd`

subroutine zgbbrd (character VECT, integer M, integer N, integer NCC,
    integer KL, integer KU, complex*16, dimension( ldab, * ) AB, integer
    LDAB, double precision, dimension( * ) D, double precision, dimension( *
    ) E, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16,
    dimension( ldpt, * ) PT, integer LDPT, complex*16, dimension( ldc, * )
    C, integer LDC, complex*16, dimension( * ) WORK, double precision,
    dimension( * ) RWORK, integer INFO)

ZGBBRD

Purpose

ZGBBRD reduces a complex general m-by-n band matrix A to real upper bidiagonal form B by a unitary transformation: Q**H * A * P = B. literally transforming your whole matrix into something slimmer, periodt 🎀

The routine computes B, and optionally forms Q or P**H, or computes Q**H*C for a given matrix C.

Parameters

VECT — VECT is CHARACTER*1. Specifies whether or not the matrices Q and P**H are to be formed.
- = 'N': do not form Q or P**H;
- = 'Q': form Q only;
- = 'P': form P**H only;
- = 'B': form both.
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.
NCC — NCC is INTEGER. The number of columns of the matrix C. NCC >= 0.
KL — KL is INTEGER. The number of subdiagonals of the matrix A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals of the matrix A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). On entry, the m-by-n band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

On exit, A is overwritten by values generated during the reduction.

LDAB — LDAB is INTEGER. The leading dimension of the array A. LDAB >= KL+KU+1.
D — D is DOUBLE PRECISION array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B.
E — E is DOUBLE PRECISION array, dimension (min(M,N)-1). The superdiagonal elements of the bidiagonal matrix B.
Q — Q is COMPLEX*16 array, dimension (LDQ,M). If VECT = 'Q' or 'B', the m-by-m unitary matrix Q. If VECT = 'N' or 'P', the array Q is not referenced.
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
PT — PT is COMPLEX*16 array, dimension (LDPT,N). If VECT = 'P' or 'B', the n-by-n unitary matrix P'. If VECT = 'N' or 'Q', the array PT is not referenced.
LDPT — LDPT is INTEGER. The leading dimension of the array PT. LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
C — C is COMPLEX*16 array, dimension (LDC,NCC). On entry, an m-by-ncc matrix C. On exit, C is overwritten by Q**H*C. C is not referenced if NCC = 0.
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
WORK — WORK is COMPLEX*16 array, dimension (max(M,N))
RWORK — RWORK is DOUBLE PRECISION 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.

💅 subroutine `zgbcon`

subroutine zgbcon (character NORM, integer N, integer KL, integer KU,
    complex*16, dimension( ldab, * ) AB, integer LDAB, integer, dimension( *
    ) IPIV, double precision ANORM, double precision RCOND, complex*16,
    dimension( * ) WORK, double precision, dimension( * ) RWORK, integer
    INFO)

ZGBCON

Purpose

ZGBCON estimates the reciprocal of the condition number of a complex general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by ZGBTRF. it's the vibe check for how sketchy your matrix is 💀

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.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). Details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix was interchanged with row IPIV(i).
ANORM — ANORM is DOUBLE PRECISION. 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 DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK — WORK is COMPLEX*16 array, dimension (2*N)
RWORK — RWORK is DOUBLE PRECISION 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.

💅 subroutine `zgbequ`

subroutine zgbequ (integer M, integer N, integer KL, integer KU, complex*16,
    dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * )
    R, double precision, dimension( * ) C, double precision ROWCND, double
    precision COLCND, double precision AMAX, integer INFO)

ZGBEQU

Purpose

ZGBEQU computes row and column scalings intended to equilibrate an M-by-N band 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. balancing your matrix so nothing is screaming louder than everything else ✨

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

Parameters

M — …
M — INTEGER. The number of rows of the matrix A. M >= 0. (no negative rows bestie, that's not a thing 💀)
N — INTEGER. The number of columns of the matrix A. N >= 0.
KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — COMPLEX*16 array, dimension (LDAB,N). The band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1. (respect the leading dimension, she's sensitive ✨)
R — DOUBLE PRECISION array, dimension (M). If INFO = 0, or INFO > M, R contains the row scale factors for A.
C — DOUBLE PRECISION array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — DOUBLE PRECISION. 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 — DOUBLE PRECISION. 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 — DOUBLE PRECISION. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. (overflow and underflow are the two toxic exes of floating point, we do not let them in the group chat 🙈)
INFO — 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.

💅 subroutine `zgbequb` (integer M, integer N, integer KL, integer KU, complex16, dimension( ldab, ) AB, integer LDAB, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)

ZGBEQUB

✨ Purpose:

ZGBEQUB 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. ✨ glow-up for your matrix, no cap ✨

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 ZGEEQU 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). (she's giving "still cute, just different vibe" 🦋)

🌸 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.
KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — COMPLEX*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

LDAB — INTEGER. The leading dimension of the array A. LDAB >= max(1,M).
R — DOUBLE PRECISION array, dimension (M). If INFO = 0 or INFO > M, R contains the row scale factors for A.
C — DOUBLE PRECISION array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — DOUBLE PRECISION. 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 — DOUBLE PRECISION. 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 — DOUBLE PRECISION. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
INFO — 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.

💅 subroutine `zgbrfs` (character TRANS, integer N, integer KL, integer KU, integer NRHS, complex16, dimension( ldab, ) AB, integer LDAB, complex16, dimension( ldafb, ) AFB, integer LDAFB, integer, dimension( * ) IPIV, complex16, dimension( ldb, ) B, integer LDB, complex16, dimension( ldx, ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex16, dimension( ) WORK, double precision, dimension( * ) RWORK, integer INFO)

ZGBRFS

✨ Purpose:

ZGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution. (she's the "let me double check that for you bestie" friend and honestly we stan 💖)

🌸 Parameters

TRANS — CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
N — INTEGER. The order of the matrix A. N >= 0.
KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AB — COMPLEX*16 array, dimension (LDAB,N). The original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB — COMPLEX*16 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
LDAFB — INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
IPIV — INTEGER array, dimension (N). The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). (yes queen, we swapped those rows, it's called pivoting and it's iconic 🎀)
B — COMPLEX*16 array, dimension (LDB,NRHS). The right hand side matrix B.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — COMPLEX*16 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by ZGBTRS. On exit, the improved solution matrix X. (she came in mid, she's leaving upgraded 💅)
LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
FERR — DOUBLE PRECISION 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. (she's dramatic, she rounds up, but she's never lying to you fr fr)
BERR — DOUBLE PRECISION array, dimension (NRHS). 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).
WORK — COMPLEX*16 array, dimension (2*N)
RWORK — DOUBLE PRECISION array, dimension (N)
INFO — INTEGER:
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

🔥 Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Not a parameter you pass in, bestie — it's internal. She handles her own business. 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zgbrfsx 💖

okay bestie, here's the whole glow-up — signature first, no edits, no cap:

   subroutine zgbrfsx (character TRANS, character EQUED, integer N, integer KL,
       integer KU, integer NRHS, complex*16, dimension( ldab, * ) AB, integer
       LDAB, complex*16, dimension( ldafb, * ) AFB, integer LDAFB, integer,
       dimension( * ) IPIV, double precision, dimension( * ) R, double
       precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer
       LDB, complex*16, dimension( ldx , * ) X, integer LDX, double precision
       RCOND, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double
       precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision,
       dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision,
       dimension( * ) PARAMS, complex*16, dimension( * ) WORK, double
       precision, dimension( * ) RWORK, integer INFO)
       ZGBRFSX

Purpose ✨

So what does ZGBRFSX actually do, you ask? It takes your computed solution to a system of linear equations and levels it up — plus it hands you error bounds and backward error estimates for that solution. And not just the normwise error bound either: this one gives you the maximum componentwise error bound too, if it can. The deets on all those bounds live in the comments for ERR_BNDS_NORM and ERR_BNDS_COMP. 👑

Heads up though: your original system of linear equations might have already been equilibrated before you called this routine — that's exactly what the EQUED, R and C arguments are telling you about. And if that happened? The solution and error bounds that come back are for the original, unequilibrated system. Periodt.

Also, a lil bundle of optional parameters is hanging out inside the PARAMS array. Those settings decide how the refinement gets done, but honestly the defaults are usually giving exactly what you need. If the defaults are fine by you, you can just pass NPARAMS = 0 and that stops the source code from ever touching the PARAMS argument. Frugal. Efficient. Iconic. 💅

Parameters 🎀

TRANS — TRANS is CHARACTER*1, and it says which form of the system of equations we're working with:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
EQUED — EQUED is CHARACTER*1 and it tells you what kind of equilibration got done to A before this routine was called. You genuinely need this, because it's how the solution and error bounds get computed correctly:
- = '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.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 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.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). It's your original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A lives in the j-th column of the array AB like so: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N). All the deets of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. 🫶
LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from ZGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
R — R is DOUBLE PRECISION 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. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. If R is output, each element of R is a power of the radix. If R is input, each element of R should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
C — C is DOUBLE PRECISION 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. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. If C is output, each element of C is a power of the radix. 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.
B — B is COMPLEX*16 array, dimension (LDB,NRHS). The right hand side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX*16 array, dimension (LDX,NRHS). On entry: the solution matrix X, as computed by ZGETRS. 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).
RCOND — RCOND is DOUBLE PRECISION. Reciprocal scaled condition number — an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that happened). If this is less than the machine precision (and especially if it's zero), then the matrix is singular to working precision. 💀 But wait — note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. Don't let the vibe check alone ruin your day.
BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). Componentwise relative backward error. That's 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. The 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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array is carrying info about various error bounds and condition numbers for 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, described below. There are currently 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array is carrying info about various error bounds and condition numbers for 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 (because that's what the componentwise relative error depends on), and the type of error information described below. There are currently 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. And if N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

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

The second index in ERR_BNDS_COMP(:,err) holds these three fields, and bestie they are not interchangeable, ok:

err = 1 — the 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
err = 2 — the 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 👀
err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

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

NPARAMS

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

PARAMS

PARAMS is DOUBLE PRECISION array, dimension NPARAMS

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

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not.
- Default: 1.0D+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. ⚠️
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (attempt componentwise convergence)

WORK

WORK is COMPLEX*16 array, dimension (2*N)

RWORK

RWORK is DOUBLE PRECISION array, dimension (2*N)

INFO

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is guaranteed. 🎉
< 0: If INFO = -i, the *i*th argument had an illegal value.
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
= N+J: The solution corresponding to the *J*th 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 *J*th 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 *J*th 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 zgbtf2` 💅

subroutine zgbtf2 (integer M, integer N, integer KL, integer KU, complex*16,
    dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV,
    integer INFO)

ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. (unblocked = the chill, no-frills, Level 2 BLAS one, iykyk)

Purpose

ZGBTF2 computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.

This is the unblocked version of the algorithm, calling Level 2 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.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N).
- On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The *j*th column of A is stored in the *j*th column of the array AB as follows:

AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
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.

Further Details

The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges.

🔥 `subroutine zgbtrf` 🔥

subroutine zgbtrf (integer M, integer N, integer KL, integer KU, complex*16,
    dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV,
    integer INFO)

ZGBTRF ✨

Purpose

ZGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS. (yes queen, blocked = faster, main character 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.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N).
- On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The *j*th column of A is stored in the *j*th column of the array AB as follows:

AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
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.
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.

bestie this one is IMPORTANT 😭 that > 0 case means the factorization finished, it just handed you a U that's exactly singular — so the second you try to solve a system with it, you're dividing by zero. Not a vibe. 💀

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details

Okay so the band storage scheme? It's giving ✨blueprint✨. Here's the example, for when M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine, periodt. Elements marked + don't need to be set on entry, but the routine does need them — they're where it stashes elements of U because of fill-in that comes from the row interchanges. 🫶

subroutine zgbtrs (character TRANS, integer N, integer KL, integer KU,
    integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB,
    integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer
    LDB, integer INFO)

🌈 ZGBTRS 🌈

Purpose

ZGBTRS solves a system of linear equations

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

with a general band matrix A, using the LU factorization that ZGBTRF already computed. 🎀

Parameters

TRANS — TRANS is CHARACTER*1, and it specifies the form of the system of equations, no cap:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
N — N is INTEGER. The order of the matrix A. N >= 0.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). It holds the details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization live in rows KL+KU+2 to 2*KL+KU+1. 🦋
LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix was interchanged with row IPIV(i).
B — B is COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. glow up ✨
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= 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 zggbak (character JOB, character SIDE, integer N, integer ILO,
    integer IHI, double precision, dimension( * ) LSCALE, double precision,
    dimension( * ) RSCALE, integer M, complex*16, dimension( ldv, * ) V,
    integer LDV, integer INFO)

💜 ZGGBAK 💜

Purpose

ZGGBAK forms the right or left eigenvectors of a complex generalized eigenvalue problem A*x = lambda*B*x, by doing a backward transformation on the computed eigenvectors of the balanced pair of matrices that ZGGBAL output. it's literally the un-balancing step, iykyk 💅

Parameters

JOB — JOB is CHARACTER*1. Specifies the type of backward transformation required:
- = 'N': do nothing, return immediately;
- = 'P': do backward transformation for permutation only;
- = 'S': do backward transformation for scaling only;
- = 'B': do backward transformations for both permutation and scaling.

JOB must be the same as the argument JOB supplied to ZGGBAL. no switching up on us bestie 🙈

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 ZGGBAL. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
LSCALE — LSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and/or scaling factors applied to the left side of A and B, as returned by ZGGBAL.
RSCALE — RSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and/or scaling factors applied to the right side of A and B, as returned by ZGGBAL.
M — M is INTEGER. The number of columns of the matrix V. M >= 0.
V — V is COMPLEX*16 array, dimension (LDV,M). On entry, the matrix of right or left eigenvectors to be transformed, as returned by ZTGEVC. On exit, V is overwritten by the transformed eigenvectors.
LDV — LDV is INTEGER. The leading dimension of the matrix 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.

Further Details

See R.C. Ward, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. 📚

subroutine zggbal (character JOB, integer N, complex*16, dimension( lda, * )
    A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer
    ILO, integer IHI, double precision, dimension( * ) LSCALE, double
    precision, dimension( * ) RSCALE, double precision, dimension( * ) WORK,
    integer INFO)

🧚 ZGGBAL 🧚

Purpose

ZGGBAL balances a pair of general complex matrices (A,B). And what does that even mean? Two steps, bestie:

First, it permutes A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal. Second, it applies a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. ✨

Balancing may reduce the 1-norm of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. it's giving self care for your matrices 🌸

Parameters

JOB — JOB is CHARACTER*1. Specifies the operations to be performed on A and B:
- = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(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 matrices A and B. N >= 0.
A — A is COMPLEX*16 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.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimension (LDB,N). On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= 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 and B(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.
LSCALE — LSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then

  LSCALE(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.

RSCALE — RSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then

  RSCALE(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.

WORK — WORK is DOUBLE PRECISION array, dimension (lwork). lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'.
INFO
INFO is INTEGER — bestie, here's the tea on the exit code 🫶
- = 0: it worked, we ate, successful exit 💅
- < 0: if INFO = -i, then the i-th argument was giving illegal-value energy, and we do NOT stan 🙈

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

🦋 subroutine zla_gbamv 🦋

subroutine zla_gbamv (integer TRANS, integer M, integer N, integer KL,
    integer KU, double precision ALPHA, complex*16, dimension( ldab, * ) AB,
    integer LDAB, complex*16, dimension( * ) X, integer INCX, double
    precision BETA, double precision, dimension( * ) Y, integer INCY)

ZLA_GBAMV performs a matrix-vector operation to calculate error bounds.

Purpose:

ZLA_GBAMV 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. Slay.

This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. A zero entry is considered 'symbolic' if all multiplications involved in computing that entry have at least one zero multiplicand. No cap, that's literally what it does 💀

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.

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.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
ALPHA — ALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
AB — AB is COMPLEX*16 array, dimension ( LDAB, n ). Before entry, the leading m by n part of the array AB must contain the matrix of coefficients. Unchanged on exit.
LDAB — LDAB is INTEGER. On entry, LDAB specifies the first dimension of AB as declared in the calling (sub) program. LDAB must be at least max( 1, m ). Unchanged on exit.
X — X is COMPLEX*16 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 DOUBLE PRECISION. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.
Y — Y is DOUBLE PRECISION array, dimension ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. She's getting rewritten, periodt 💅
INCY — INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

Level 2 Blas routine.

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💜 double precision function zla_gbrcond_c 💜

double precision function zla_gbrcond_c (character TRANS, integer N, integer
    KL, integer KU, complex*16, dimension( ldab, * ) AB, integer LDAB,
    complex*16, dimension( ldafb, * ) AFB, integer LDAFB, integer,
    dimension( * ) IPIV, double precision, dimension( * ) C, logical CAPPLY,
    integer INFO, complex*16, dimension( * ) WORK, double precision,
    dimension( * ) RWORK)

ZLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices. Iconic behaviour, honestly 🌈

Purpose:

ZLA_GBRCOND_C Computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.

Parameters

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate Transpose = Transpose)
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i).
C — C is DOUBLE PRECISION array, dimension (N). The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — CAPPLY is LOGICAL. If .TRUE. then access the vector C in the formula above.
INFO — INFO is INTEGER.
- = 0: Successful exit. 🎀
- i > 0: The i-th argument is invalid. 🚩
WORK — WORK is COMPLEX*16 array, dimension (2*N). Workspace.
RWORK — RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 double precision function zla_gbrcond_x 🌸

double precision function zla_gbrcond_x (character TRANS, integer N, integer
    KL, integer KU, complex*16, dimension( ldab, * ) AB, integer LDAB,
    complex*16, dimension( ldafb, * ) AFB, integer LDAFB, integer,
    dimension( * ) IPIV, complex*16, dimension( * ) X, integer INFO,
    complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)

ZLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices. Obsessed with her 🦋

Purpose:

ZLA_GBRCOND_X Computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX*16 vector.

Parameters

TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate Transpose = Transpose)
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i).
X — X is COMPLEX*16 array, dimension (N). The vector X in the formula op(A) * diag(X).
INFO — INFO is INTEGER.
- = 0: Successful exit. ✨
- i > 0: The i-th argument is invalid. 🚩
WORK — WORK is COMPLEX*16 array, dimension (2*N). Workspace.
RWORK —
RWORK is a DOUBLE PRECISION array, dimension (N). It's the workspace, bestie — not the star of the show, but the show does not happen without it. 💜

🎀 Author

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

✨ `zla_gbrfsx_extended` ✨

okay so this is the big one, the extra-precise iterative refinement girlie for general banded matrices. Here's the whole signature, no cuts, no edits:

subroutine zla_gbrfsx_extended (integer PREC_TYPE, integer TRANS_TYPE,
    integer N, integer KL, integer KU, integer NRHS, complex*16, dimension(
    ldab, * ) AB, integer LDAB, complex*16, dimension( ldafb, * ) AFB,
    integer LDAFB, integer, dimension( * ) IPIV, logical COLEQU, double
    precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer
    LDB, complex*16, dimension( ldy, * ) Y, integer LDY, double precision,
    dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension(
    nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * )
    ERR_BNDS_COMP, complex*16, dimension( * ) RES, double precision,
    dimension(*) AYB, complex*16, dimension( * ) DY, complex*16, dimension(
    * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision
    RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)

ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. 🫶

🌸 Purpose

ZLA_GBRFSX_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 ZGBRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.

So like, the tea here: this routine is called by ZGBRFSX to do the iterative refinement, and it is only on the hook for setting those second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. That's the whole job. Don't ask it for more. 💅

🦋 Parameters

`PREC_TYPE`

PREC_TYPE is INTEGER. It says which intermediate precision we're using during refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P is one of:

'S': Single
'D': Double
'I': Indigenous
'X' or 'E': Extra

`TRANS_TYPE`

TRANS_TYPE is INTEGER. It says which transposition operation we're doing on A. The value is defined by ILATRANS(T) where T is a CHARACTER and T is one of:

'N': No transpose
'T': Transpose
'C': Conjugate transpose

`N`

N is INTEGER. The number of linear equations, i.e., the order of the matrix A. Constraint: N >= 0. ✨

`KL`

KL is INTEGER. The number of subdiagonals within the band of A. Constraint: KL >= 0.

`KU`

KU is INTEGER. The number of superdiagonals within the band of A. Constraint: KU >= 0.

`NRHS`

NRHS is INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.

`AB`

AB is COMPLEX*16 array, dimension (LDAB,N). On entry, the N-by-N matrix A. 💖

`LDAB`

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

`AFB`

AFB is COMPLEX*16 array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by ZGBTRF.

`LDAFB`

LDAFB is INTEGER. The leading dimension of the array AF. Constraint: LDAF >= max(1,N).

yes bestie, it literally says LDAF there and not LDAFB. we are reproducing it exactly, do not come for me 🙈

`IPIV`

IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i).

`COLEQU`

COLEQU is LOGICAL. If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.

`C`

C is DOUBLE PRECISION 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.

🚨 lowkey important: powers of the radix only, or the error estimates stop being trustworthy. the vibes must stay clean.

`B`

B is COMPLEX*16 array, dimension (LDB,NRHS). The right-hand-side matrix B.

`LDB`

LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N).

`Y`

Y is COMPLEX*16 array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by ZGBTRS. On exit, the improved solution matrix Y. 👑

`LDY`

LDY is INTEGER. The leading dimension of the array Y. Constraint: LDY >= max(1,N).

`BERR_OUT`

BERR_OUT is DOUBLE PRECISION 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 ZLA_LIN_BERR.

`N_NORMS`

N_NORMS is INTEGER. Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP).

If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.

`ERR_BNDS_NORM`

ERR_BNDS_NORM is DOUBLE PRECISION 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.

This subroutine is only responsible for setting the second field above.

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

`ERR_BNDS_COMP`

ERR_BNDS_COMP is DOUBLE PRECISION 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 — 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 — Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field above.

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

`RES`

RES is COMPLEX*16 array, dimension (N). Workspace to hold the intermediate residual.

`AYB`

AYB — it's a DOUBLE PRECISION array, dimension (N). Just workspace, bestie. ✨
DY — COMPLEX*16 array, dimension (N). Workspace to hold the intermediate solution. 💖
Y_TAIL — COMPLEX*16 array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. 🦋
RCOND — DOUBLE PRECISION. 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. 😭
ITHRESH — 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 ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. 💅
RTHRESH — DOUBLE PRECISION. 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 — DOUBLE PRECISION. 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 — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 🙈
INFO — INTEGER.
- = 0: Successful exit.
- < 0: if INFO = -i, the ith argument to ZGBTRS had an illegal value. 💀

✨ Author ✨

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

`double precision function zla_gbrpvgrw`

double precision function zla_gbrpvgrw (integer N, integer KL, integer KU, integer NCOLS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldafb, * ) AFB, integer LDAFB)

ZLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.

Purpose

ZLA_GBRPVGRW 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. 😬

Parameters

N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
NCOLS — INTEGER. The number of columns of the matrix A. NCOLS >= 0.
AB — COMPLEX*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB — COMPLEX*16 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
LDAFB — INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

Author

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

`subroutine zungbr`

subroutine zungbr (character VECT, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)

ZUNGBR

Purpose

ZUNGBR generates one of the complex unitary matrices Q or P**H determined by ZGEBRD when reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q and P**H are defined as products of elementary reflectors H(i) or G(i) respectively.

If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q is of order M:

if m >= k, Q = H(1) H(2) . . . H(k) and ZUNGBR returns the first n columns of Q, where m >= n >= k;
if m < k, Q = H(1) H(2) . . . H(m-1) and ZUNGBR returns Q as an M-by-M matrix.

If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H is of order N:

if k < n, P**H = G(k) . . . G(2) G(1) and ZUNGBR returns the first m rows of P**H, where n >= m >= k;
if k >= n, P**H = G(n-1) . . . G(2) G(1) and ZUNGBR returns P**H as an N-by-N matrix.

Parameters

VECT — CHARACTER*1. Specifies whether the matrix Q or the matrix P**H is required, as defined in the transformation applied by ZGEBRD:
- = 'Q': generate Q;
- = 'P': generate P**H.
M — INTEGER. The number of rows of the matrix Q or P**H to be returned. M >= 0.
N — INTEGER. The number of columns of the matrix Q or P**H to be returned. N >= 0. If VECT = 'Q', M >= N >= min(M,K); if VECT = 'P', N >= M >= min(N,K).
K — INTEGER. If VECT = 'Q', the number of columns in the original M-by-K matrix reduced by ZGEBRD. If VECT = 'P', the number of rows in the original K-by-N matrix reduced by ZGEBRD. K >= 0.
A — COMPLEX*16 array, dimension (LDA,N). On entry, the vectors which define the elementary reflectors, as returned by ZGEBRD. On exit, the M-by-N matrix Q or P**H.
LDA — INTEGER. The leading dimension of the array A. LDA >= M.
TAU — COMPLEX*16 array, dimension (min(M,K)) if VECT = 'Q', (min(N,K)) if VECT = 'P'. TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P**H, as returned by ZGEBRD in its array argument TAUQ or TAUP.
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,min(M,N)). For optimum performance LWORK >= min(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 — 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.

Author

Generated automatically by Doxygen for LAPACK from the source code.

This page is from LAPACK version 3.11.0, dated Sun Nov 27 2022. Stay slay, bestie! 💖✨

📦 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

complex16GBcomputational(3)

💖 complex16GBcomputational (3) — LAPACK 🎀

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions

🦋 DETAILED DESCRIPTION 🦋

👑 FUNCTION DOCUMENTATION 👑

💅 subroutine zgbbrd

Purpose

Parameters

Author

💅 subroutine zgbcon

Purpose

Parameters

Author

💅 subroutine zgbequ

Purpose

Parameters

👑 Author

💅 subroutine zgbequb (integer M, integer N, integer KL, integer KU, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)

✨ Purpose:

🌸 Parameters

👑 Author

✨ Purpose:

🌸 Parameters

🔥 Internal Parameters:

👑 Author

subroutine zgbrfsx 💖

Purpose ✨

Parameters 🎀

NPARAMS

PARAMS

WORK

RWORK

INFO

🎀 Author 🎀

💅 subroutine zgbtf2 💅

Purpose

Parameters

Author

Further Details

🔥 subroutine zgbtrf 🔥

Purpose

Parameters

Author

Further Details

🌈 ZGBTRS 🌈

Purpose

Parameters

Author

💜 ZGGBAK 💜

Purpose

Parameters

Author

Further Details

🧚 ZGGBAL 🧚

Purpose

Parameters

Author 👑

Further Details:

🦋 subroutine zla_gbamv 🦋

Purpose:

Parameters

Author 👑

💜 double precision function zla_gbrcond_c 💜

Purpose:

Parameters

Author 👑

🌸 double precision function zla_gbrcond_x 🌸

Purpose:

Parameters

🎀 Author

✨ zla_gbrfsx_extended ✨

🌸 Purpose

🦋 Parameters

PREC_TYPE

TRANS_TYPE

N

KL

KU

💖 `complex16GBcomputational` (3) — LAPACK 🎀

💅 subroutine `zgbbrd`

💅 subroutine `zgbcon`

💅 subroutine `zgbequ`

💅 subroutine `zgbequb` (integer M, integer N, integer KL, integer KU, complex16, dimension( ldab, ) AB, integer LDAB, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)

💅 `subroutine zgbtf2` 💅

🔥 `subroutine zgbtrf` 🔥

✨ `zla_gbrfsx_extended` ✨

`PREC_TYPE`

`TRANS_TYPE`

`N`

`KL`

`KU`

`NRHS`

`AB`

`LDAB`

`AFB`

`LDAFB`

`IPIV`

`COLEQU`

`C`

`B`

`LDB`

`Y`

`LDY`

`BERR_OUT`

`N_NORMS`

`ERR_BNDS_NORM`

`ERR_BNDS_COMP`

`RES`

`AYB`

`double precision function zla_gbrpvgrw`

`subroutine zungbr`