Developer Reference for Intel® oneAPI Math Kernel Library for C
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?gbtrs
Solves a system of linear equations with an LU-factored band coefficient matrix, with multiple right-hand sides.
Syntax
lapack_int LAPACKE_sgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const float * ab , lapack_int ldab , const lapack_int * ipiv , float * b , lapack_int ldb );
lapack_int LAPACKE_dgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const double * ab , lapack_int ldab , const lapack_int * ipiv , double * b , lapack_int ldb );
lapack_int LAPACKE_cgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const lapack_complex_float * ab , lapack_int ldab , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );
lapack_int LAPACKE_zgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int kl , lapack_int ku , lapack_int nrhs , const lapack_complex_double * ab , lapack_int ldab , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );
Include Files
- mkl.h
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
matrix_layout |
Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR). |
trans |
Must be 'N' or 'T' or 'C'. |
n |
The order of A; the number of rows in B; n≥ 0. |
kl |
The number of subdiagonals within the band of A; kl≥ 0. |
ku |
The number of superdiagonals within the band of A; ku≥ 0. |
nrhs |
The number of right-hand sides; nrhs≥ 0. |
ab |
Array ab size max(1, ldab*n) The array ab contains elements of the LU factors of the matrix A as returned by gbtrf. |
b |
Array b size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout. The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. |
ldab |
The leading dimension of the array ab; ldab≥ 2*kl + ku +1. |
ldb |
The leading dimension of b; ldb≥ max(1, n) for column major layout and ldb≥nrhs for row major layout. |
ipiv |
Array, size at least max(1, n). The ipiv array, as returned by ?gbtrf. |
Output Parameters
b |
Overwritten by the solution matrix X. |
Return Values
This function returns a value info.
If info=0, the execution is successful.
If info = -i, parameter i had an illegal value.
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.