?gbsvx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gbsvx

Computes the solution to the real or complex system of linear equations with a band coefficient matrix A and multiple right-hand sides, and provides error bounds on the solution.

Syntax

call sgbsvx( fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call dgbsvx( fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call cgbsvx( fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call 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 )

call gbsvx( ab, b, x [,kl] [,afb] [,ipiv] [,fact] [,trans] [,equed] [,r] [,c] [,ferr] [,berr] [,rcond] [,rpvgrw] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine uses the LU factorization to compute the solution to a real or 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, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

Error bounds on the solution and a condition estimate are also provided.

The routine ?gbsvx performs the following steps:

If fact = 'E', real scaling factors r and c 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 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.

Input Parameters

fact	CHARACTER*1. Must be 'F', 'N', or 'E'. Specifies whether 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. If `fact = 'F'`: on entry, `afb` and `ipiv` contain the factored form of `A`. If `equed` is not 'N', the matrix `A` is equilibrated with scaling factors given by `r` and `c`. `ab`, `afb`, and `ipiv` are not modified. If `fact = 'N'`, the matrix `A` will be copied to `afb` and factored. If `fact = 'E'`, the matrix `A` will be equilibrated if necessary, then copied to `afb` and factored.
trans	CHARACTER1. Must be 'N', 'T', or 'C'. Specifies the form of the system of equations: If `trans = 'N'`, the system has the form `AX = B` (No transpose). If `trans = 'T'`, the system has the form `A`^T`X` = `B` (Transpose). If `trans = 'C'`, the system has the form `A`^H`X` = `B` (Transpose for real flavors, conjugate transpose for complex flavors).
n	INTEGER. The number of linear equations, 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, the number of columns of the matrices `B` and `X`; `nrhs`≥ 0.
ab, afb, b, work	REAL for sgbsvx DOUBLE PRECISION for dgbsvx COMPLEX for cgbsvx DOUBLE COMPLEX for zgbsvx. Arrays: `ab(ldab,)`, `afb(ldafb,)`, `b(size ldb by )`, `work()`. The array `ab` contains the matrix `A` in band storage (see Matrix Storage Schemes). The second dimension of `ab` must be at least `max(1, n)`. If `fact = 'F'` and `equed` is not 'N', then `A` must have been equilibrated by the scaling factors in `r` and/or `c`. The array `afb` is an input argument if `fact = 'F'`. The second dimension of `afb` must be at least `max(1,n)`. It contains the factored form of the matrix `A`, that is, the factors `L` and `U` from the factorization `A = PLU` as computed by ?gbtrf. `U` is stored as an upper triangular band matrix with `kl` + `ku` superdiagonals in the first 1 + `kl` + `ku` rows of `afb`. The multipliers used during the factorization are stored in the next `kl` rows. If `equed` is not 'N', then `afb` is the factored form of the equilibrated matrix `A`. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations. The second dimension of `b` must be at least `max(1, nrhs)`. `work()` is a workspace array. The dimension of `work` must be at least `max(1,3n)` for real flavors, and at least `max(1,2*n)` for complex flavors.
ldab	INTEGER. The leading dimension of `ab`; `ldab`≥`kl`+`ku`+1.
ldafb	INTEGER. The leading dimension of `afb`; `ldafb`≥ 2*`kl`+`ku`+1.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.
ipiv	INTEGER. Array, size at least `max(1, n)`. The array `ipiv` is an input argument if `fact = 'F'`. It contains the pivot indices from the factorization `A = PLU` as computed by ?gbtrf; row `i` of the matrix was interchanged with row `ipiv`(`i`).
equed	CHARACTER1. Must be 'N', 'R', 'C', or 'B'. `equed` is an input argument if `fact = 'F'`. It specifies the form of equilibration that was done: If `equed = 'N'`, no equilibration was done (always true if `fact = 'N'`). If `equed = 'R'`, row equilibration was done, that is, `A` has been premultiplied by `diag(r)`. If `equed = 'C'`, column equilibration was done, that is, `A` has been postmultiplied by `diag(c)`. if `equed = 'B'`, both row and column equilibration was done, that is, `A` has been replaced by `diag(r)A*diag(c)`.
r, c	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays: `r` (size `n`), `c (size n)`. The array `r` contains the row scale factors for `A`, and the array `c` contains the column scale factors for `A`. These arrays are input arguments if `fact = 'F'` only; otherwise they are output arguments. If `equed = 'R'`or 'B', `A` is multiplied on the left by `diag(r)`; if `equed = 'N'` or 'C', `r` is not accessed. If `fact = 'F'` and `equed = 'R'` or 'B', each element of `r` must be positive. If `equed = 'C'`or 'B', `A` is multiplied on the right by `diag(c)`; if `equed = 'N'`or 'R', `c` is not accessed. If `fact = 'F'` and `equed = 'C'`or 'B', each element of `c` must be positive.
ldx	INTEGER. The leading dimension of the output array `x`; `ldx≥ max(1, n)`.
iwork	INTEGER. Workspace array, size at least `max(1, n)`; used in real flavors only.
rwork	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Workspace array, size at least `max(1, n)`; used in complex flavors only.

Output Parameters

x	REAL for sgbsvx DOUBLE PRECISION for dgbsvx COMPLEX for cgbsvx DOUBLE COMPLEX for zgbsvx. Array, size `ldx` by . If `info = 0` or `info = n+1`, the array `x` contains the solution matrix `X` to the original* system of equations. Note that `A` and `B` are modified on exit if `equed≠'N'`, and the solution to the equilibrated system is: `inv(diag(c))X`, if `trans = 'N'` and `equed = 'C'`or 'B'; `inv(diag(r))X`, if `trans = 'T'` or 'C' and `equed = 'R'` or 'B'. The second dimension of `x` must be at least `max(1,nrhs)`.
ab	Array `ab` is not modified on exit if `fact = 'F'` or 'N', or if `fact = 'E'` and `equed` = 'N'. If `equed≠'N'`, `A` is scaled on exit as follows: `equed` = 'R': `A = diag(r)A` `equed` = 'C': `A = Adiag(c)` `equed` = 'B': `A = diag(r)Adiag(c)`.
afb	If `fact = 'N'` or 'E', then `afb` is an output argument and on exit returns details of the `LU` factorization of the original matrix `A` (if `fact = 'N'`) or of the equilibrated matrix `A` (if `fact = 'E'`). See the description of `ab` for the form of the equilibrated matrix.
b	Overwritten by `diag(r)b` if `trans = 'N'` and `equed = 'R'` or `'B'`; overwritten by `diag(c)b` if `trans = 'T'` or `'C'` and `equed = 'C'` or `'B'`; not changed if `equed` = 'N'.
r, c	These arrays are output arguments if `fact≠'F'`. See the description of `r`, `c` in Input Arguments section.
rcond	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. An 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains the component-wise relative backward error for each solution vector `x(j)`, that is, the smallest relative change in any element of `A` or `B` that makes `x(j)` an exact solution.
ipiv	If `fact = 'N'` or 'E', 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 = 'N'`) or of the equilibrated matrix `A` (if `fact = 'E'`).
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, and `i≤n`, then `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. If `info = i`, and `i = n+1`, then `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.
equed	If `fact≠'F'`, then `equed` is an output argument. It specifies the form of equilibration that was done (see the description of `equed` in Input Arguments section).
work, rwork	On exit, `work(1)` for real flavors, or `rwork(1)` for complex flavors, contains the reciprocal pivot growth factor norm(`A`)/norm(`U`). The "max absolute element" norm is used. If `work(1)` for real flavors, or `rwork(1)` for complex flavors 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 `work(1)` for real flavors, or `rwork(1)` for complex flavors contains the reciprocal pivot growth factor for the leading `info` columns of `A`.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, and `i≤n`, then `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. If `info = i`, and `i = n+1`, then `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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gbsvx interface are as follows:

ab	Holds the array `A` of size (`kl+ku+1,n`).
b	Holds the matrix `B` of size (`n,nrhs`).
x	Holds the matrix `X` of size (`n,nrhs`).
afb	Holds the array `AF` of size (`2*kl+ku+1,n`).
ipiv	Holds the vector of length `n`.
r	Holds the vector of length `n`. Default value for each element is `r(i) = 1.0_WP`.
c	Holds the vector of length `n`. Default value for each element is `c(i) = 1.0_WP`.
ferr	Holds the vector of length (`nrhs`).
berr	Holds the vector of length (`nrhs`).
trans	Must be 'N', 'C', or 'T'. The default value is 'N'.
equed	Must be 'N', 'B', 'C', or 'R'. The default value is 'N'.
fact	Must be 'N', 'E', or 'F'. The default value is 'N'. If `fact = 'F'`, then both arguments `af` and `ipiv` must be present; otherwise, an error is returned.
rpvgrw	Real value that contains the reciprocal pivot growth factor norm(`A`)/norm(`U`).
kl	If omitted, assumed `kl = ku`.
ku	Restored as `ku = lda-kl-1`.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gbsvx

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