?hpsvx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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:

If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*U^H or A = L*D*L^H, 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.
If some d_i,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.
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

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	CHARACTER1. 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 `UDU^H`. If `uplo = 'L'`, the array `ap` stores the lower triangular part of the Hermitian matrix `A`, and `A` is factored as `LD*L^H`.
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 = UDU^H` or `A = LDL^H` 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(1,2n)`.
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 `d_ii` 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 `i≤n`, then `d_ii` 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.

Parent topic: LAPACK Linear Equation Driver Routines

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