Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 7/13/2023
Public

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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, AT*X = B, or AH*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:

  1. 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').

  2. 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.

  3. If some Ui,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.

  4. The system of equations is solved for X using the factored form of A.

  5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

  6. 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

CHARACTER*1. Must be 'N', 'T', or 'C'.

Specifies the form of the system of equations:

If trans = 'N', the system has the form A*X = B (No transpose).

If trans = 'T', the system has the form AT*X = B (Transpose).

If trans = 'C', the system has the form AH*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 = P*L*U 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(1,3*n) for real flavors, and at least max(1,2*n) for complex flavors.

ldab

INTEGER. The leading dimension of ab; ldabkl+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 = P*L*U as computed by ?gbtrf; row i of the matrix was interchanged with row ipiv(i).

equed

CHARACTER*1. 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 = A*diag(c)

equed = 'B': A = diag(r)*A*diag(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 in, 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 < infon, 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 in, 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.