?ptsvx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?ptsvx

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

Syntax

lapack_int LAPACKE_sptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs, const float* d, const float* e, float* df, float* ef, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_dptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs, const double* d, const double* e, double* df, double* ef, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );

lapack_int LAPACKE_cptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs, const float* d, const lapack_complex_float* e, float* df, lapack_complex_float* ef, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_zptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs, const double* d, const lapack_complex_double* e, double* df, lapack_complex_double* ef, const 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 = L*D*L^T (real)/A = L*D*L^H (complex) 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 tridiagonal 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 ?ptsvx performs the following steps:

If fact = 'N', the matrix A is factored as A = L*D*L^T (real flavors)/A = L*D*L^H (complex flavors), where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U^T*D*U (real flavors)/A = U^H*D*U (complex flavors).
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.

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' or 'N'. Specifies whether or not the factored form of the matrix `A` is supplied on entry. If `fact = 'F'`: on entry, `df` and `ef` contain the factored form of `A`. Arrays `d`, `e`, `df`, and `ef` will not be modified. If `fact = 'N'`, the matrix `A` will be copied to `df` and `ef`, and factored.
n	The order of matrix `A`; `n`≥ 0.
nrhs	The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
d, df	Arrays: `d` (size `n`), `df` (size `n`). The array `d` contains the `n` diagonal elements of the tridiagonal matrix `A`. The array `df` is an input argument if `fact = 'F'` and on entry contains the `n` diagonal elements of the diagonal matrix `D` from the `LDL^T` (real)/ `LDL^H` (complex) factorization of `A`.
e,ef,b	Arrays: `e` (size `n` -1), `ef` (size `n` -1), `b`, size max(ldbnrhs) for column major layout and max(ldbn) for row major layout. The array `e` contains the `(n - 1)` subdiagonal elements of the tridiagonal matrix `A`. The array `ef` is an input argument if `fact = 'F'` and on entry contains the `(n - 1)` subdiagonal elements of the unit bidiagonal factor `L` from the `LDL^T` (real)/ `LDL^H` (complex) factorization of `A`. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations.
ldb	The leading dimension of `b`; `ldb`≥ max(1, n) for column major layout and `ldb`≥nrhs for row major layout.
ldx	The leading dimension of `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 system of equations.
df, ef	These arrays are output arguments if `fact = 'N'`. See the description of `df`, `ef` 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` 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.

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

?ptsvx

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

See Also