Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 3/31/2023
Public

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?hpsvx

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

Syntax

call chpsvx( fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call zhpsvx( fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call hpsvx( ap, b, x [,uplo] [,afp] [,ipiv] [,fact] [,ferr] [,berr] [,rcond] [,info] )

Include Files
  • mkl.fi, lapack.f90
Description

The routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A*X = B, where A is a n-by-n Hermitian 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 ?hpsvx 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*UH or A = L*D*LH, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is a Hermitian 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

fact

CHARACTER*1. 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

CHARACTER*1. 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 Hermitian matrix A, and A is factored as U*D*UH.

If uplo = 'L', the array ap stores the lower triangular part of the Hermitian matrix A, and A is factored as L*D*LH.

n

INTEGER. The order of matrix A; n 0.

nrhs

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

ap, afp, b, work

COMPLEX for chpsvx

DOUBLE COMPLEX for zhpsvx.

Arrays: ap(size *), afp(size *), b(size ldb by *), work(*).

The array ap contains the upper or lower triangle of the Hermitian 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*UH or A = L*D*LH as computed by ?hptrf, 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.

work(*) is a workspace array.

The dimension of arrays ap and afp must be at least max(1,n(n+1)/2); the second dimension of b must be at least max(1,nrhs); the dimension of work must be at least max(1,2*n).

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

ipiv

INTEGER.

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 ?hptrf.

If ipiv(i) = 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-1)-th row and column of A was interchanged with the m-th row and column.

If uplo = 'L'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+1)-th row and column of A was interchanged with the m-th row and column.

ldx

INTEGER. The leading dimension of the output array x; ldx max(1, n).

rwork

REAL for chpsvx

DOUBLE PRECISION for zhpsvx.

Workspace array, size at least max(1, n).

Output Parameters

x

COMPLEX for chpsvx

DOUBLE COMPLEX for zhpsvx.

Array, size ldx by *.

If info = 0 or info = n+1, the array x contains the solution matrix X to the system of equations. The second dimension of x must be at least max(1,nrhs).

afp, ipiv

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

rcond

REAL for chpsvx

DOUBLE PRECISION for zhpsvx.

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

REAL for chpsvx

DOUBLE PRECISION for zhpsvx.

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

REAL for chpsvx

DOUBLE PRECISION for zhpsvx.

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.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter 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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine hpsvx interface are as follows:

ap

Holds the array A of size (n*(n+1)/2).

b

Holds the matrix B of size (n,nrhs).

x

Holds the matrix X of size (n,nrhs).

afp

Holds the array AF of size (n*(n+1)/2).

ipiv

Holds the vector with the number of elements n.

ferr

Holds the vector with the number of elements nrhs.

berr

Holds the vector with the number of elements nrhs.

uplo

Must be 'U' or 'L'. The default value is 'U'.

fact

Must be 'N' or 'F'. The default value is 'N'. If fact = 'F', then both arguments af and ipiv must be present; otherwise, an error is returned.