Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a real or complex symmetric coefficient matrix A stored in packed format, and provides error bounds on the solution.

Syntax

lapack_int LAPACKE_sspsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const float* ap, float* afp, lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_dspsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const double* ap, double* afp, lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );

lapack_int LAPACKE_cspsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* ap, lapack_complex_float* afp, lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );

lapack_int LAPACKE_zspsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* ap, lapack_complex_double* afp, lapack_int* ipiv, 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 diagonal pivoting factorization to compute the solution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric matrix stored in packed format, 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 ?spsvx performs the following steps:

  1. If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*UT orA = L*D*LT, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  2. If some di,i= 0, so that D is exactly singular, 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.

  3. The system of equations is solved for X using the factored form of A.

  4. 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 has been supplied on entry.

If fact = 'F': on entry, afp and ipiv contain the factored form of A. Arrays ap, afp, and ipiv are not modified.

If fact = 'N', the matrix A is copied to afp and factored.

uplo

Must be 'U' or 'L'.

Indicates whether the upper or lower triangular part of A is stored and how A is factored:

If uplo = 'U', the array ap stores the upper triangular part of the symmetric matrix A, and A is factored as U*D*UT.

If uplo = 'L', the array ap stores the lower triangular part of the symmetric matrix A; A is factored as L*D*LT.

n

The order of matrix A; n 0.

nrhs

The number of right-hand sides, the number of columns in B; nrhs 0.

ap, afp, b

Arrays: ap (size max(1,n*(n+1)/2), afp (size max(1,n*(n+1)/2), bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout.

The array ap contains the upper or lower triangle of the symmetric matrix A in packed storage (see Matrix Storage Schemes).

The array afp is an input argument if fact = 'F'. It contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*UT or A = L*D*LT as computed by ?sptrf, in the same storage format as 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 ldbnrhs for row major layout.

ipiv

Array, size at least max(1, n). The array ipiv is an input argument if fact = 'F'. It contains details of the interchanges and the block structure of D, as determined by ?sptrf.

If ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row and column of A was interchanged with the k-th row and column.

If uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and i-th row and column of A was interchanged with the m-th row and column.

If uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column.

ldx

The leading dimension of the output array x; ldx max(1, n) for column major layout and ldxnrhs for row major layout.

Output Parameters

x

Array, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n) for row major layout.

If info = 0 or info = n+1, the array x contains the solution matrix X to the system of equations.

afp, ipiv

These arrays are output arguments if fact = 'N'. See the description of afp, ipiv in Input Arguments section.

rcond

An estimate of the reciprocal condition number of the matrix A. 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, berr

Arrays, size at least max(1, nrhs). Contain the component-wise forward and relative backward errors, respectively, for each solution vector.

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 in, then dii is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.

If info = i, and i = n + 1, then D 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.

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