complex16GBsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 complex16GBsolve(3) 💖 the LAPACK band-solve girlies

okay so. this is a page from the LAPACK manual, and it's about the complex16 group of solve driver functions for GB matrices. bestie, if your matrix is banded and complex, this is your era. 🦋

✨ NAME ✨

complex16GBsolve — complex16

🎀 SYNOPSIS 🎀

Functions

subroutine zgbsv (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
     ZGBSV computes the solution to system of linear equations A * X = B
    for GB matrices (simple driver)
subroutine zgbsvx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
    IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
    INFO)
     ZGBSVX computes the solution to system of linear equations A * X =
    B for GB matrices
subroutine zgbsvxx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
    IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
    ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
     ZGBSVXX computes the solution to system of linear equations A * X =
    B for GB matrices

🌈 Detailed Description 🌈

This is the whole squad 💅 — the group of complex16 solve driver functions for GB matrices. three functions, one mission: solve that banded system.

👑 Function Documentation 👑

🌸 subroutine zgbsv 🌸

subroutine zgbsv (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)

ZGBSV computes the solution to system of linear equations A * X = B for GB matrices — and it's the simple driver, so it's the low-key, no-frills one. ✨

💜 Purpose

ZGBSV computes the solution to a complex system of linear equations

A * X = B

where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

A = L * U

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. That factored form of A then gets used to solve the system of equations A * X = B. periodt. 💅

🍓 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.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
AB — 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(N,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 — INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
IPIV — INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i).
B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. 🫶
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed. 💀

🧚 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. iconic behaviour tbh. 🔥

🦋 subroutine zgbsvx 🦋

subroutine zgbsvx (character FACT, character TRANS, 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, character EQUED, 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( * ) FERR, double
    precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double
    precision, dimension( * ) RWORK, integer INFO)

ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices. this one's the extra one — it does error bounds and a condition estimate too. 💖

💜 Purpose

ZGBSVX uses the LU factorization to compute the solution to a complex system of linear equations

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

where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. she's thorough. 💅

🌸 Description:

The following steps are performed by this subroutine:

If FACT = 'E', real scaling factors are computed to equilibrate the system:

TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = L * U,

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.

If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

🍓 Parameters

FACT — CHARACTER*1. Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
- = 'F': On entry, AFB and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not modified.
- = 'N': The matrix A will be copied to AFB and factored.
- = 'E': The matrix A will be equilibrated if necessary, then copied to AFB and factored.
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 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.
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). 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)

If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).

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

AFB — okay bestie, this one is a COMPLEX*16 array, dimension (LDAFB,N). Here's the whole vibe:

If FACT = 'F', then AFB is an input argument and on entry contains 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. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.
If FACT = 'N', then AFB is an output argument and on exit returns details of the LU factorization of A.
If FACT = 'E', then AFB is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of AB for the form of the equilibrated matrix).

LDAFB — is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. Slay, keep that dimension in check. 💅

IPIV — is INTEGER array, dimension (N).

If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.

EQUED — is CHARACTER*1. Specifies the form of equilibration that was done.

= 'N': No equilibration (always true if FACT = 'N').
= '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).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

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.

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.

B — is COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit,

if EQUED = 'N', B is not modified;
if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.

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

X — is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

LDX — is INTEGER. The leading dimension of the array X. LDX >= max(1,N).

RCOND — is DOUBLE PRECISION. The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.

FERR — is 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.

BERR — is 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 — is COMPLEX*16 array, dimension (2*N).

RWORK — is DOUBLE PRECISION array, dimension (N). On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

INFO — is INTEGER.

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
- <= N: U(i,i) 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+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

✨ Author ✨

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

✨ `subroutine zgbsvxx` ✨

   subroutine zgbsvxx (character FACT, character TRANS, 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, character EQUED, 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 RPVGRW, 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)

ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices. 💅

🎀 Purpose 🎀

ZGBSVXX uses the LU factorization to compute the solution to a complex*16 system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds are returned. ZGBSVXX will return a solution with a tiny guaranteed error (O(eps) where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers also are calculated and returned.

ZGBSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous ZGBSVXX call will also produce a solution with either O(eps) errors or warnings, but we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what ZGBSVXX would itself produce.

🌸 Description 🌸

The following steps are performed:

If FACT = 'E', double precision scaling factors are computed to equilibrate the system:

TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

Okay so if some U(i,i)=0, meaning U is exactly singular — like fully, no coming back from that — the routine just returns with INFO = i. 💀 Otherwise, the factored form of A gets used to estimate the condition number of the matrix A (go peep the RCOND argument for that whole situation). And if the reciprocal of the condition number is less than machine precision? She still goes on to solve for X and compute error bounds exactly like it says below. Committed queen. 👑
The system of equations is solved for X using the factored form of A. Slay, that's it, that's the step. ✨
By default — that's unless PARAMS(LA_LINRX_ITREF_I) is set to zero — the routine uses iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision, which is giving overachiever, honestly. 🎀
If equilibration was used, the matrix X gets premultiplied by diag(C) (if TRANS = 'N') or by diag(R) (if TRANS = 'T' or 'C'), so that it solves the original system before equilibration. So the math works out the way you actually asked for it. 💖

Some optional parameters are bundled together in the PARAMS array — think of it as a lil settings squad. These settings determine how refinement is performed, but ngl the defaults are often completely acceptable. And if the defaults are acceptable to you, you can pass NPARAMS = 0, which stops the source code from ever accessing the PARAMS argument. Efficiency. 💅

✨ Parameters ✨

FACT — FACT is CHARACTER*1. Specifies whether or not the factored form of the matrix A is supplied on entry, and if it isn't, whether the matrix A should be equilibrated before it gets factored.
- = 'F': On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. A, AF, and IPIV are not modified.
- = 'N': The matrix A will be copied to AF and factored.
- = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
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, aka the order of the matrix A. Requirement: N >= 0. 🫶
KL — KL is INTEGER. The number of subdiagonals within the band of A. Requirement: KL >= 0.
KU — KU is INTEGER. The number of superdiagonals within the band of A. Requirement: KU >= 0.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e. the number of columns of the matrices B and X. Requirement: NRHS >= 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 like this:

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

If FACT = 'F' and EQUED is not 'N', then AB must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled like this:

EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).

LDAB — LDAB is INTEGER. The leading dimension of the array AB. Requirement: LDAB >= KL+KU+1.
AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N).
- If FACT = 'F', then AFB is an input argument and on entry it contains 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. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.
- If FACT = 'N', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the original matrix A.
- If FACT = 'E', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for what the equilibrated matrix looks like).
LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. Requirement: LDAFB >= 2*KL+KU+1.
IPIV — IPIV is INTEGER array, dimension (N).
- If FACT = 'F', then IPIV is an input argument and on entry it contains the pivot indices from the factorization A = P*L*U as computed by ZGETRF; row i of the matrix was interchanged with row IPIV(i).
- If FACT = 'N', then IPIV is an output argument and on exit it contains the pivot indices from the factorization A = P*L*U of the original matrix A.
- If FACT = 'E', then IPIV is an output argument and on exit it contains the pivot indices from the factorization A = P*L*U of the equilibrated matrix A.
EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done. 👑
- = 'N': No equilibration (always true if FACT = 'N').
- = '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).

EQUED is an input argument if FACT = 'F'; otherwise it's an output argument.

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', 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 ensure a reliable solution and error estimates. 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 is not equivalent to the input matrix, which produces 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', 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 to ensure a reliable solution and error estimates. 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 is not equivalent to the input matrix, which produces error estimates that may not be reliable. Same energy as R, fr fr. 💀

B — B is COMPLEX*16 array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit:
- if EQUED = 'N', B is not modified;
- if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
- if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
LDB — LDB is INTEGER. The leading dimension of the array B. Requirement: LDB >= max(1,N).
X — X is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Heads up bestie: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
LDX — LDX is INTEGER. The leading dimension of the array X. Requirement: LDX >= max(1,N).
RCOND — RCOND is DOUBLE PRECISION. Reciprocal scaled condition number. This is 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), the matrix is singular to working precision. But note — the error may still be small even if this number is very small and the matrix looks ill-conditioned. Don't panic-judge her. 😭
RPVGRW — RPVGRW is DOUBLE PRECISION. Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is what's used here. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. Which also means the solution X, the estimated condition numbers, and the error bounds could be unreliable. If the factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. In ZGESVX, this quantity is returned in RWORK(1).
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. Iconic. ⭐
N_ERR_BNDS — N_ERR_BNDS is INTEGER. The number of error bounds to return for each right hand side and for each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. 👀
ERR_BNDS_NORM

`ERR_BNDS_NORM` 💖

ERR_BNDS_NORM is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains info 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) * 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 a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains info 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) * 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 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 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 extra-precise refinement algorithm. (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 Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

Author 💖

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

Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole thing, bestie! This page is from LAPACK version 3.11.0, dated Sun Nov 27 2022, for complex16GBsolve(3). Periodt. 💅✨

📦 where this page comes from

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

complex16GBsolve(3)

💖 complex16GBsolve(3) 💖 the LAPACK band-solve girlies

✨ NAME ✨

🎀 SYNOPSIS 🎀

Functions

🌈 Detailed Description 🌈

👑 Function Documentation 👑

🌸 subroutine zgbsv 🌸

💜 Purpose

🍓 Parameters

🧚 Author

🌟 Further Details:

🦋 subroutine zgbsvx 🦋

💜 Purpose

🌸 Description:

🍓 Parameters

✨ Author ✨

✨ subroutine zgbsvxx ✨

🎀 Purpose 🎀

🌸 Description 🌸

✨ Parameters ✨

ERR_BNDS_NORM 💖

ERR_BNDS_COMP 🦋

NPARAMS 🎀

PARAMS ✨

WORK 💅

RWORK 🌸

INFO 👑

Author 💖

Author 🌈

✨ `subroutine zgbsvxx` ✨

`ERR_BNDS_NORM` 💖

`ERR_BNDS_COMP` 🦋

`NPARAMS` 🎀

`PARAMS` ✨

`WORK` 💅

`RWORK` 🌸

`INFO` 👑