Visible to Intel only — GUID: GUID-EF70810C-6FB0-44B9-B471-47AFBDC028E8
Visible to Intel only — GUID: GUID-EF70810C-6FB0-44B9-B471-47AFBDC028E8
?gbrfs
Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.
Syntax
call sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )
call dgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info )
call cgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )
call zgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info )
call gbrfs( ab, afb, ipiv, b, x [,kl] [,trans] [,ferr] [,berr] [,info] )
Include Files
- mkl.fi, lapack.f90
Description
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X = B or AH*X = B with a band matrix A, with multiple right-hand sides. For each computed solution vector x, the routine computes the component-wise backward errorβ. This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:
|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).
Finally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||x||∞ (here xe is the exact solution).
Before calling this routine:
Input Parameters
trans |
CHARACTER*1. Must be 'N' or 'T' or 'C'. Indicates the form of the equations: If trans = 'N', the system has the form A*X = B. If trans = 'T', the system has the form AT*X = B. If trans = 'C', the system has the form AH*X = B. |
n |
INTEGER. The order of the matrix A; n≥ 0. |
kl |
INTEGER. The number of sub-diagonals within the band of A; kl≥ 0. |
ku |
INTEGER. The number of super-diagonals within the band of A; ku≥ 0. |
nrhs |
INTEGER. The number of right-hand sides; nrhs≥ 0. |
ab,afb,b,x,work |
REAL for sgbrfs DOUBLE PRECISION for dgbrfs COMPLEX for cgbrfs DOUBLE COMPLEX for zgbrfs. Arrays: ab(size ldab by *) contains the original band matrix A, as supplied to ?gbtrf, but stored in rows from 1 to kl + ku + 1. afb(size ldafb by *) contains the factored band matrix A, as returned by ?gbtrf. b(size ldb by *) contains the right-hand side matrix B. x(size ldx by *) contains the solution matrix X. work(*) is a workspace array. The second dimension of ab and afb must be at least max(1, n); the second dimension of b and x must be at least max(1, nrhs); the dimension of work must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors. |
ldab |
INTEGER. The leading dimension of ab. |
ldafb |
INTEGER. The leading dimension of afb. |
ldb |
INTEGER. The leading dimension of b; ldb≥ max(1, n). |
ldx |
INTEGER. The leading dimension of x; ldx≥ max(1, n). |
ipiv |
INTEGER. Array, size at least max(1, n). The ipiv array, as returned by ?gbtrf. |
iwork |
INTEGER. Workspace array, size at least max(1, n). |
rwork |
REAL for cgbrfs DOUBLE PRECISION for zgbrfs. Workspace array, size at least max(1, n). |
Output Parameters
x |
The refined solution matrix X. |
ferr, berr |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays, size at least max(1, nrhs). Contain the component-wise forward and backward errors, respectively, for each solution vector. |
info |
INTEGER. If info =0, the execution is successful. If info = -i, the i-th parameter had an illegal value. |
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 gbrfs interface are as follows:
ab |
Holds the array A of size (kl+ku+1,n). |
afb |
Holds the array AF of size (2*kl*ku+1,n). |
ipiv |
Holds the vector of length n. |
b |
Holds the matrix B of size (n, nrhs). |
x |
Holds the matrix X of size (n, nrhs). |
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'. |
kl |
If omitted, assumed kl = ku. |
ku |
Restored as ku = lda-kl-1. |
Application Notes
The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.
For each right-hand side, computation of the backward error involves a minimum of 4n(kl + ku) floating-point operations (for real flavors) or 16n(kl + ku) operations (for complex flavors). In addition, each step of iterative refinement involves 2n(4kl + 3ku) operations (for real flavors) or 8n(4kl + 3ku) operations (for complex flavors); the number of iterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point operations for real flavors or 8n2 for complex flavors.