Solves a system of linear equations with an LU-factored band matrix, with multiple right-hand sides.
Syntax
FORTRAN 77:
call sgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )
call dgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )
call cgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )
call zgbtrs( trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info )
FORTRAN 95:
call gbtrs( ab, b, ipiv, [, kl] [, trans] [, info] )
C:
lapack_int LAPACKE_<?>gbtrs( int matrix_order, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const <datatype>* ab, lapack_int ldab, const lapack_int* ipiv, <datatype>* b, lapack_int ldb );
Description
The routine solves for X the following systems of linear equations:
A*X = B |
if trans='N', |
AT*X = B |
if trans='T', |
AH*X = B |
if trans='C' (for complex matrices only). |
Here A is an LU-factored general band matrix of order n with kl non-zero subdiagonals and ku nonzero superdiagonals. Before calling this routine, call ?gbtrf to compute the LU factorization of A.
Input Parameters
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
trans |
CHARACTER*1. Must be 'N' or 'T' or 'C'. |
n |
INTEGER. The order of A; the number of rows in B; 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; nrhs ≥ 0. |
ab, b |
REAL for sgbtrs DOUBLE PRECISION for dgbtrs COMPLEX for cgbtrs DOUBLE COMPLEX for zgbtrs. Arrays: ab(ldab,*), b(ldb,*). The array ab contains the matrix A in band storage (see Matrix Storage Schemes). The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of ab must be at least max(1, n), and the second dimension of b at least max(1,nrhs). |
ldab |
INTEGER. The leading dimension of the array ab; ldab ≥ 2*kl + ku +1. |
ldb |
INTEGER. The leading dimension of b; ldb ≥ max(1, n). |
ipiv |
INTEGER. Array, DIMENSION at least max(1, n). The ipiv array, as returned by ?gbtrf. |
Output Parameters
b |
Overwritten by the solution matrix X. |
info |
INTEGER. If info=0, the execution is successful. If info = -i, the i-th parameter had an illegal value. |
Fortran 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 Fortran 95 Interface Conventions.
Specific details for the routine gbtrs interface are as follows:
ab |
Holds the array A of size (2*kl+ku+1,n). |
b |
Holds the matrix B of size (n, nrhs). |
ipiv |
Holds the vector of length min(m, n). |
kl |
If omitted, assumed kl = ku. |
ku |
Restored as lda-2*kl-1. |
trans |
Must be 'N', 'C', or 'T'. The default value is 'N'. |
Application Notes
For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A + E)x = b, where
|E| ≤ c(kl + ku + 1)ε P|L||U|
c(k) is a modest linear function of k, and ε is the machine precision.
If x0 is the true solution, the computed solution x satisfies this error bound:
where cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞ ≤ ||A-1||∞ ||A||∞ = κ∞(A).
Note that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might not be equal to κ∞(A).
The approximate number of floating-point operations for one right-hand side vector is 2n(ku + 2kl) for real flavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that kl and ku are much less than min(m,n).
To estimate the condition number κ∞(A), call ?gbcon.
To refine the solution and estimate the error, call ?gbrfs.