complexGBcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

complexGBcomputational(3) 💖

okay bestie, welcome to complexGBcomputational from the LAPACK manual, section 3. this is the group chat where all the complex GB matrix computational functions hang out. it's long, so buckle up 🎀

NAME ✨

complexGBcomputational — it's giving complex. that's literally it, that's the name. 💅

SYNOPSIS 🌸

Functions 🦋

here's the whole roster, no gatekeeping:

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

CGBBRD — the icon itself.

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

CGBCON — periodt.

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

CGBEQU — yes queen.

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

CGBEQUB — her sister, equally slay.

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

CGBRFS — main character energy fr.

subroutine cgbrfsx (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)

CGBRFSX — the extended cut 🍓.

subroutine cgbtf2 (M, N, KL, KU, AB, LDAB, IPIV, INFO)

CGBTF2 — bestie, she computes the LU factorization of a general band matrix using the unblocked version of the algorithm. lowkey the blueprint.

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

CGBTRF — iconic.

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

CGBTRS — ate.

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

CGGBAK — obsessed with her.

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

CGGBAL — queen behavior.

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

CLA_GBAMV — she performs a matrix-vector operation to calculate error bounds. no cap.

real function cla_gbrcond_c (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, C, CAPPLY, INFO, WORK, RWORK)

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

real function cla_gbrcond_x (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, X, INFO, WORK, RWORK)

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

subroutine cla_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)

CLA_GBRFSX_EXTENDED — she 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. the way that she does ALL of that. 😭

real function cla_gbrpvgrw (N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)

CLA_GBRPVGRW — she computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix. ⭐

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

CUNGBR — the moment. 👑

Detailed Description 💜

okay so. this is the group of complex computational functions for GB matrices. that's the whole vibe. 🦋

Function Documentation 🎀

subroutine cgbbrd 🌸

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

CGBBRD — the one and only.

Purpose: ✨

CGBBRD 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.

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 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. (yeah, she replaces A, that's just how she moves 💅)
LDAB — LDAB is INTEGER. The leading dimension of the array A. LDAB >= KL+KU+1.
D — D is REAL array, dimension (min(M,N)). The diagonal elements of the bidiagonal matrix B.
E — E is REAL array, dimension (min(M,N)-1). The superdiagonal elements of the bidiagonal matrix B.
Q — Q is COMPLEX 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 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 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 array, dimension (max(M,N))
RWORK — RWORK 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.

subroutine cgbcon 🦋

subroutine cgbcon (character NORM, integer N, integer KL, integer KU, complex, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CGBCON — her.

Purpose: ✨

CGBCON 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 CGBTRF.

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 array, dimension (LDAB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 REAL. If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK — WORK is COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL array, dimension (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 cgbequ 🌈

subroutine cgbequ (integer M, integer N, integer KL, integer KU, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)

CGBEQU — the equilibration queen.

Purpose: ✨

CGBEQU 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.

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. (translation: bestie, she tries, and honestly she usually eats, but it's not a promise 💅)

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 — INTEGER energy. This is how many subdiagonals live inside the band of A. Rule of the land: KL >= 0.
KU — also INTEGER. The number of superdiagonals chillin' within the band of A. KU >= 0, bestie, no exceptions.
AB — COMPLEX array, dimension (LDAB,N). This is the band matrix A itself, stored in rows 1 to KL+KU+1. The j-th column of A gets parked 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(m,j+kl).

LDAB — INTEGER. The leading dimension of the array AB. Gotta be LDAB >= KL+KU+1.
R — REAL array, dimension (M). If INFO = 0, or INFO > M, then R is holding the row scale factors for A. 🫶
C — REAL array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — REAL. If INFO = 0 or INFO > M, ROWCND holds the ratio of the smallest R(i) to the largest R(i). And fr fr: if ROWCND >= 0.1 and AMAX is neither too big nor too tiny, it is not worth scaling by R.
COLCND — REAL. If INFO = 0, COLCND holds the ratio of the smallest C(i) to the largest C(i). If COLCND >= 0.1, it is not worth scaling by C. periodt.
AMAX — REAL. The absolute value of the largest matrix element. If AMAX is super close to overflow or super close to underflow, then bestie… the matrix should be scaled. 😭
INFO — INTEGER. The vibe check:
- = 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.

🌸 `cgbequb`

subroutine cgbequb (integer M, integer N, integer KL, integer KU, complex,
    dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real,
    dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)

CGBEQUB — say it with me now 💅

💖 Purpose

CGBEQUB 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. Heads up though: use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice. 🦋

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

✨ 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 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 — REAL array, dimension (M). If INFO = 0 or INFO > M, R contains the row scale factors for A.
C — REAL array, dimension (N). If INFO = 0, C contains the column scale factors for A.
ROWCND — 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 — 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 — 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 — 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.

🔥 `cgbrfs`

subroutine cgbrfs (character TRANS, integer N, integer KL, integer KU,
    integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex,
    dimension( ldafb, * ) AFB, integer LDAFB, integer, dimension( * ) IPIV,
    complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, *
    ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR,
    complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CGBRFS — glow up your solution, bestie 💅✨

💜 Purpose

CGBRFS 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 thorough, she's precise, she's doing the most and we love her for it 🫶

🌈 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 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 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 CGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
B — COMPLEX 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 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by CGBTRS. On exit, the improved solution matrix X. ✨ the glow up ✨
LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (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 a little dramatic, that's fine.
BERR — REAL 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 array, dimension (2*N)
RWORK — REAL 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.

⭐ Author

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

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

Purpose 🎀

CGBRFSX is out here doing the most: it takes your already-computed solution to a linear system and makes it better, then hands you error bounds and backward error estimates for that solution. Bestie, it's giving glow-up. ✨

Not only does it do the normwise error bound thing, it also gives you the maximum componentwise error bound when it can. If you wanna know the deets on those bounds, go peek at the comments for ERR_BNDS_NORM and ERR_BNDS_COMP — they spill all the tea. 🫖

Here's the part you actually need to clock: the original system you're solving might have been equilibrated before you called this routine. That's what the EQUED, R, and C arguments are telling it. And if that's the case, then the solution and error bounds it hands back are for the original, unequilibrated system — not the scaled one. Iconic behavior honestly. 💅

Some optional parameters are bundled up in the PARAMS array. These settings decide how the refinement gets done, but ngl, the defaults are usually totally fine. And if the defaults are fine for you? Just pass NPARAMS = 0 and the source code won't even touch the PARAMS argument. No cap, it just skips it. 🙈

Parameters 👑

TRANS — TRANS is CHARACTER*1, and it says which form of the system we're dealing with:
- = 'N': A * X = B (no transpose, the chill one)
- = 'T': A**T * X = B (transpose)
- = 'C': A**H * X = B (conjugate transpose)
EQUED — EQUED is CHARACTER*1, telling you what kind of equilibration got done to A before you called this routine. This matters, because without it the solution and error bounds can't be computed correctly. So like, don't fumble this one. 🫶
- = 'N': no equilibration at all, clean slate
- = 'R': row equilibration — A got premultiplied by diag(R)
- = 'C': column equilibration — A got postmultiplied by diag(C)
- = 'B': both, bestie! A was replaced by diag(R) * A * diag(C), and the right hand side B was changed accordingly
N — N is INTEGER, the order of the matrix A. N >= 0.
KL — KL is INTEGER, how many subdiagonals live inside the band of A. KL >= 0.
KU — KU is INTEGER, how many superdiagonals live inside the band of A. KU >= 0.
NRHS — NRHS is INTEGER, the number of right hand sides, aka the number of columns in B and X. NRHS >= 0.
AB — AB is COMPLEX array, dimension (LDAB,N). This is your 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 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 array, dimension (LDAFB,N). All the details of the LU factorization of the band matrix A, exactly as computed by CGBTRF. Here's the layout, keep up: 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. 🧚
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 CGETRF; for 1<=i<=N, row i of the matrix got swapped with row IPIV(i). Shuffling the deck, basically. 🔥
R — R is REAL array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', then A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', then R doesn't get accessed at all, it's just vibing. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', then every 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 make sure you get a reliable solution and reliable error estimates. Scaling by powers of the radix doesn't cause rounding errors — unless the result underflows or overflows, that is. 😭 And rounding errors during scaling lead to refining with a matrix that isn't equivalent to the input matrix, which produces error estimates that may not be reliable. So like, be careful.
C — C is REAL array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', then A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', then C doesn't get accessed at all. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', every 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 so your solution and error estimates are actually reliable. Scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that isn't equivalent to the input matrix, and then your error estimates may not be reliable. Same energy as R, fr fr.
B — B is COMPLEX array, dimension (LDB,NRHS). The right hand side matrix B. Simple as that. 💖
LDB — LDB is INTEGER, the leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX array, dimension (LDX,NRHS). On entry, this is the solution matrix X, as computed by CGETRS. On exit, it's the improved solution matrix X. Main character transformation arc, honestly. ✨
LDX — LDX is INTEGER, the leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. The 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 heads up: the error can still be small even when this number is tiny and the matrix looks ill-conditioned.
BERR — BERR is REAL 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 would make X(j) an exact solution.
N_ERR_BNDS — N_ERR_BNDS is INTEGER. How many 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 holds info about various error bounds and condition numbers corresponding to the normwise relative error, which is defined like this:

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 info 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 holds info about various error bounds and condition numbers corresponding to the componentwise relative error, defined like this:

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

The array is indexed by the right-hand side i (which is what the componentwise relative error depends on) and the type of error information, described below. There are currently up to three pieces of info returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. 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.

and it's giving three whole fields, so let's break it down bestie 💅

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, 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. no cap 🫶
err = 3 — the reciprocal condition number. This is the estimated componentwise reciprocal condition number. It gets 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.

And how is that Z built, you ask? 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 behavior honestly ✨

See Lapack Working Note 165 for further details and extra cautions. Read it fr fr, there's lore there 📚

🌸 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. (so like, it doesn't even get a glance 👀)

🎀 PARAMS 🎀

PARAMS is REAL array, dimension NPARAMS

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

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. ⚠️ like, proceed with caution bestie
PARAMS(LA_LINRX_CWISE_I = 3) — flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (attempt componentwise convergence)

🦋 WORK 🦋

WORK is COMPLEX array, dimension (2*N)

🌈 RWORK 🌈

RWORK is REAL array, dimension (2*N)

⭐ INFO ⭐

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is guaranteed. 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. (one report only, the algorithm is not a gossip 💅)

👑 Author 👑

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

🧚 subroutine cgbtf2 🧚

subroutine cgbtf2 (integer M, integer N, integer KL, integer KU, complex,
    dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV,
    integer INFO)

CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. (the unblocked one, the scrappy little sibling 🫶)

✨ Purpose ✨

CGBTF2 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 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. 😭 danger

🌸 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 cgbtrf 🔥

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

CGBTRF

✨ Purpose ✨

CGBTRF 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. (glow up unlocked ✨)

💜 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 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 💖

and bestie, this one keeps going — next part's got the rest 👑

INFO — INTEGER bestie, here's the whole vibe:
- = 0: successful exit, we love that for us 💅
- < 0: if INFO = -i, the i-th argument had an illegal value, so something's sus 🙈
- > 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. NOT the vibe, bestie — division by zero is a crime scene.

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:

             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 cgbtrs ✨

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

CGBTRS

Purpose:

CGBTRS 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 computed
by CGBTRF.

Parameters

TRANS — CHARACTER*1 bestie, this tells us the form of the system. It can be:
- '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. Periodt.
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 matrix B. NRHS >= 0.
AB — COMPLEX array, dimension (LDAB,N) details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 — INTEGER the leading dimension of the array AB. LDAB >= 2*KL+KU+1.
IPIV — INTEGER array, dimension (N) the pivot indices; for 1 <= i <= N, row i of the matrix was interchanged with row IPIV(i).
B — COMPLEX array, dimension (LDB,NRHS) on entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — INTEGER the leading dimension of the array B. LDB >= max(1,N).
INFO — INTEGER exit status, bestie:
- = 0: successful exit, we ate 💅
- < 0: if INFO = -i, the i-th argument had an illegal value, so fix it 🙈

Author

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

✨ subroutine cggbak ✨

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

CGGBAK

Purpose:

CGGBAK forms the right or left eigenvectors of a complex generalized
eigenvalue problem A*x = lambda*B*x, by backward transformation on
the computed eigenvectors of the balanced pair of matrices output by
CGGBAL.

Parameters

JOB — 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 CGGBAL.
SIDE — CHARACTER*1:
- 'R': V contains right eigenvectors;
- 'L': V contains left eigenvectors.
N — INTEGER the number of rows of the matrix V. N >= 0.
ILO — INTEGER bestie, that's it.
IHI — INTEGER the integers ILO and IHI determined by CGGBAL. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
LSCALE — REAL array, dimension (N) details of the permutations and/or scaling factors applied to the left side of A and B, as returned by CGGBAL.
RSCALE — REAL array, dimension (N) details of the permutations and/or scaling factors applied to the right side of A and B, as returned by CGGBAL.
M — INTEGER the number of columns of the matrix V. M >= 0.
V — COMPLEX array, dimension (LDV,M) on entry, the matrix of right or left eigenvectors to be transformed, as returned by CTGEVC. On exit, V is overwritten by the transformed eigenvectors.
LDV — INTEGER the leading dimension of the matrix V. LDV >= max(1,N).
INFO — INTEGER exit status:
- = 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 cggbal ✨

subroutine cggbal (character JOB, integer N, complex, dimension( lda, * ) A,
    integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer ILO,
    integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE,
    real, dimension( * ) WORK, integer INFO)

CGGBAL

Purpose:

CGGBAL balances a pair of general complex matrices (A,B).  This
involves, first, permuting 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; 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 matrices, and improve the
accuracy of the computed eigenvalues and/or eigenvectors in the
generalized eigenvalue problem A*x = lambda*B*x.

Parameters

JOB — 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 — INTEGER the order of the matrices A and B. N >= 0.
A — COMPLEX array, dimension (LDA,N) on entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced.
LDA — INTEGER the leading dimension of the array A. LDA >= max(1,N).
B — COMPLEX 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 — INTEGER the leading dimension of the array B. LDB >= max(1,N).
ILO — INTEGER bestie, that's it.
IHI — 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 — REAL 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 — REAL 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 — REAL 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 — INTEGER exit status:
- = 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.
NAG Ltd.

Further Details 💡

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

Like, if you wanna go deeper bestie, the receipts are right there 📚

subroutine `cla_gbamv` 🦋

subroutine cla_gbamv (integer TRANS, integer M, integer N, integer KL,
    integer KU, real ALPHA, complex, dimension( ldab, * ) AB, integer LDAB,
    complex, dimension( * ) X, integer INCX, real BETA, real, dimension( * )
    Y, integer INCY)

CLA_GBAMV does a matrix-vector operation to calculate error bounds. it's giving very precise, very demure 💅

Purpose ✨

so basically CLA_GBAMV performs one of these two matrix-vector operations, and you pick which one with TRANS:

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 👑

ngl this function's whole job is calculating error bounds. and here's the tea: to protect against underflow while it's evaluating, components in the resulting vector get perturbed away from zero by (N+1) times the underflow threshold 🫣 but — and this is the iconic part — to prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. a zero entry counts as 'symbolic' if all the multiplications involved in computing that entry have at least one zero multiplicand. no cap, that's the rule 📏

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 REAL. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
AB — AB is COMPLEX 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 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.
Y — Y is REAL array, dimension ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.
INCY — INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

Level 2 Blas routine. 🌈

Author 💜

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

real function `cla_gbrcond_c` 🧚

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

CLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices. she's a math girlie fr fr 📐✨

Purpose 🌸

okay so CLA_GBRCOND_C computes the infinity norm condition number of

op(A) * inv(diag(C))

where C is a REAL vector. 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)

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 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 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 CGBTRF; row i of the matrix was interchanged with row IPIV(i).
C — C is REAL array, dimension (N). The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — CAPPLY is LOGICAL. If .TRUE. then access the vector C in the formula above.
INFO — INFO is INTEGER.

= 0:  Successful exit.
i > 0:  The ith argument is invalid.

WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

Author 💜

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

real function `cla_gbrcond_x` 🦋

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

CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices. the _X version, bestie — she's the main character here 👑

Purpose 🌸

CLA_GBRCOND_X computes the infinity norm condition number of

op(A) * diag(X)

where X is a COMPLEX vector. iconic, honestly 💅

Parameters 🎀

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

= 'N':  A * X = B     (No transpose)
= 'T':  A**T * X = B  (Transpose)
= 'C':  A**H * X = B  (Conjugate Transpose = 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 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 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 CGBTRF; row i of the matrix was interchanged with row IPIV(i).
X — X is COMPLEX array, dimension (N). The vector X in the formula op(A) * diag(X).
INFO — INFO is INTEGER.

= 0:  Successful exit.
i > 0:  The ith argument is invalid.

WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

Author 💜

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
RCOND — bestie, we were doing so good and then the source text just… stops 😭 fr fr, this is exactly where the original cuts off — the parameter name RCOND shows up and then ✨nothing✨, no type, no description, no vibe. So the tea on what RCOND actually does gets spilled in the next part, no cap. Not me gatekeeping a parameter I literally cannot see 💀

Anyway — that's the edge of what this part has for you. The parameter list is still going, so hold that thought and keep scrolling 🫶

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. 💅
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 ERR_BNDS_NORM and ERR_BNDS_COMP 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. Slay. 👑
- < 0: if INFO = -i, the ith argument to CGBTRS had an illegal value. 😭

Author 🫶

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

real function `cla_gbrpvgrw` 💖

   real function cla_gbrpvgrw (integer N, integer KL, integer KU, integer
       NCOLS, complex, dimension( ldab, * ) AB, integer LDAB, complex,
       dimension( ldafb, * ) AFB, integer LDAFB)

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

Purpose ✨

CLA_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. So if this number is tiny, maybe don't trust those error bounds, bestie. 🚩

Parameters 💅

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.
NCOLS — NCOLS is INTEGER. The number of columns of the matrix A. NCOLS >= 0.
AB — AB is COMPLEX 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 array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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.

Author 🫶

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

subroutine `cungbr` 🌈

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

CUNGBR

Purpose ✨

CUNGBR generates one of the complex unitary matrices Q or P**H determined by CGEBRD 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 CUNGBR returns the first n columns of Q, where m >= n >= k;
if m < k, Q = H(1) H(2) . . . H(m-1) and CUNGBR 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 CUNGBR 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 CUNGBR returns P**H as an N-by-N matrix.

Parameters 💖

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

Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

That's the tea, bestie! This page is from LAPACK version 3.11.0, generated on Sun Nov 27 2022 for complexGBcomputational(3). Stay slay, and remember: every flag and parameter matters. 💅✨

📦 where this page comes from

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💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complexGBcomputational(3)

complexGBcomputational(3) 💖

NAME ✨

SYNOPSIS 🌸

Functions 🦋

Detailed Description 💜

Function Documentation 🎀

subroutine cgbbrd 🌸

Purpose: ✨

Parameters 🫶

Author 💖

subroutine cgbcon 🦋

Purpose: ✨

Parameters 🫶

Author 💖

subroutine cgbequ 🌈

Purpose: ✨

Parameters 🫶

👑 Author

🌸 cgbequb

💖 Purpose

✨ Parameters

🎀 Author

🔥 cgbrfs

💜 Purpose

🌈 Parameters

🧚 Internal Parameters

⭐ Author

Purpose 🎀

Parameters 👑

🌸 NPARAMS 🌸

🎀 PARAMS 🎀

🦋 WORK 🦋

🌈 RWORK 🌈

⭐ INFO ⭐

👑 Author 👑

🧚 subroutine cgbtf2 🧚

✨ Purpose ✨

💜 Parameters 💜

🌸 Author 🌸

🎀 Further Details: 🎀

🔥 subroutine cgbtrf 🔥

✨ Purpose ✨

💜 Parameters 💜

💖 INFO 💖

Author

Further Details:

✨ subroutine cgbtrs ✨

CGBTRS

Purpose:

Parameters

Author

✨ subroutine cggbak ✨

CGGBAK

Purpose:

Parameters

Author

Further Details:

✨ subroutine cggbal ✨

CGGBAL

Purpose:

Parameters

Author

Further Details 💡

subroutine cla_gbamv 🦋

Purpose ✨

Parameters 🎀

Author 💜

real function cla_gbrcond_c 🧚

Purpose 🌸

Parameters 🎀

Author 💜

real function cla_gbrcond_x 🦋

Purpose 🌸

Parameters 🎀

Author 💜

Author 🫶

real function cla_gbrpvgrw 💖

Purpose ✨

Parameters 💅

🌸 `cgbequb`

🔥 `cgbrfs`

subroutine `cla_gbamv` 🦋

real function `cla_gbrcond_c` 🧚

real function `cla_gbrcond_x` 🦋

real function `cla_gbrpvgrw` 💖

subroutine `cungbr` 🌈