?pbsvx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 6/24/2024

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?pbsvx

Uses the Cholesky factorization to compute the solution to the system of linear equations with a symmetric (Hermitian) positive-definite band coefficient matrix A, and provides error bounds on the solution.

Syntax

lapack_int LAPACKE_spbsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, float* ab, lapack_int ldab, float* afb, lapack_int ldafb, char* equed, float* s, float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_dpbsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, double* ab, lapack_int ldab, double* afb, lapack_int ldafb, char* equed, double* s, double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );

lapack_int LAPACKE_cpbsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, lapack_complex_float* ab, lapack_int ldab, lapack_complex_float* afb, lapack_int ldafb, char* equed, float* s, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_zpbsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int kd, lapack_int nrhs, lapack_complex_double* ab, lapack_int ldab, lapack_complex_double* afb, lapack_int ldafb, char* equed, double* s, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );

Include Files

mkl.h

Description

The routine uses the Cholesky factorization A=U^T*U (real flavors) / A=U^H*U (complex flavors) or A=L*L^T (real flavors) / A=L*L^H (complex flavors) to compute the solution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric or Hermitian positive definite band matrix, 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 ?pbsvx performs the following steps:

If fact = 'E', real scaling factors s are computed to equilibrate the system:
```
diag(s)*A*diag(s)*inv(diag(s))*X = diag(s)*B.
```
Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.
If fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if fact = 'E') as

A = U^T*U (real), A = U^H*U (complex), if uplo = 'U',

or A = L*L^T (real), A = L*L^H (complex), if uplo = 'L',

where U is an upper triangular band matrix and L is a lower triangular band matrix.
If the leading i-by-i principal minor is not positive definite, 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(s) so that it solves the original system before equilibration.

Input Parameters

matrix_layout	Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
fact	Must be 'F', 'N', or 'E'. Specifies whether or not 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` contains the factored form of `A`. If `equed = 'Y'`, the matrix `A` has been equilibrated with scaling factors given by `s`. `ab` and `afb` will not be 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.
uplo	Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored: If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	The order of matrix `A`; `n`≥ 0.
kd	The number of superdiagonals or subdiagonals in the matrix `A`; `kd`≥ 0.
nrhs	The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
ab, afb, b	Arrays: `ab`(size max(1, ldabn)), `afb`(size max(1, ldafbn)), `b`of size max(1, ldbnrhs) for column major layout and max(1, ldbn) for row major layout. The array `ab` contains the upper or lower triangle of the matrix `A` in band storage (see Matrix Storage Schemes). If `fact = 'F'` and `equed` = 'Y', then `ab` must contain the equilibrated matrix `diag(s)Adiag(s)`. The array `afb` is an input argument if `fact = 'F'`. It contains the triangular factor `U` or `L` from the Cholesky factorization of the band matrix `A` in the same storage format as `A`. If `equed = 'Y'`, 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.
ldab	The leading dimension of `ab`; `ldab`≥`kd`+1.
ldafb	The leading dimension of `afb`; `ldafb`≥`kd`+1.
ldb	The leading dimension of `b`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.
equed	Must be 'N' or 'Y'. `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 = 'Y'`, equilibration was done, that is, `A` has been replaced by `diag(s)Adiag(s)`.
s	Array, size (`n`). The array `s` contains the scale factors for `A`. This array is an input argument if `fact = 'F'` only; otherwise it is an output argument. If `equed = 'N'`, `s` is not accessed. If `fact = 'F'` and `equed` = 'Y', each element of `s` must be positive.
ldx	The leading dimension of the output array `x`; ldx≥ max(1, n) for column major layout and `ldx`≥nrhs for row major layout.

Output Parameters

x	Array, size max(1, ldxnrhs) for column major layout and max(1, ldxn) for row major layout. If `info` = 0 or `info = n+1`, the array `x` contains the solution matrix `X` to the original system of equations. Note that if `equed = 'Y'`, `A` and `B` are modified on exit, and the solution to the equilibrated system is `inv(diag(s))*X`.
ab	On exit, if `fact = 'E'`and `equed` = 'Y', `A` is overwritten by `diag(s)Adiag(s).`
afb	If `fact = 'N'`or 'E', then `afb` is an output argument and on exit returns the triangular factor `U` or `L` from the Cholesky factorization `A=U^TU` or `A=LL^T` (real routines), `A=U^HU` or `A=LL^H` (complex routines) 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(s)*B`, if `equed = 'Y'`; not changed if `equed` = 'N'.
s	This array is an output argument if `fact≠'F'`. See the description of `s` in Input Arguments section.
rcond	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	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-1]` 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	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.
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).

Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info = -i, parameter i had an illegal value.

If info = i, and i≤n, the leading minor of order i (and therefore the matrix A itself) is not positive definite, so the factorization could not be completed, and 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.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?pbsvx

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

See Also