Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?hprfs
Refines the solution of a system of linear equations with a packed complex Hermitian coefficient matrix and estimates the solution error.
Syntax
call chprfs ( uplo , n , nrhs , ap , afp , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zhprfs ( uplo , n , nrhs , ap , afp , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call hprfs ( ap , afp , ipiv , b , x [ , uplo ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
chprfs zhprfs hprfs
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a packed complex Hermitian matrix A , with multiple right-hand sides. For each computed solution vector x , the routine computes the component-wise backward error \(\beta\) . This error is the smallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed system:
\(|\delta a_{ij}| \leq \beta|a_{ij}|, |\delta b_{i}| \leq \beta|b_{i}|\) such that \((A + \delta A)x = (b + \delta b)\) .
Finally, the routine estimates the component-wise forward error in the computed solution \(||x - x_{e}||_{\infty}/||x||_{\infty}\) (here x_{e} is the exact solution).
Before calling this routine:
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
If uplo = 'U' , the upper triangle of A is stored.
If uplo = 'L' , the lower triangle of A is stored.
n
INTEGER . The order of the matrix A ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ap , afp , b , x , work
COMPLEX for chprfs
DOUBLE COMPLEX for zhprfs .
Arrays:
ap(size *) contains the original packed matrix A , as supplied to ?hptrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.) .
afp(size *) contains the factored packed matrix A , as returned by ?hptrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.) .
b(size ldb by *) contains the right-hand side matrix B .
x(size ldx by *) contains the solution matrix X .
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 and x 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) .
ldx
INTEGER . The leading dimension of x ; ldx≥ max(1, n) .
ipiv
INTEGER .
Array, size at least max(1, n) . The ipiv array, as returned by ?hptrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix using packed storage.) .
rwork
REAL for chprfs
DOUBLE PRECISION for zhprfs .
Workspace array, size at least max(1, n) .
Output Parameters
x
The refined solution matrix X .
ferr , berr
REAL for chprfs .
DOUBLE PRECISION for zhprfs .
Arrays, size at least max(1,nrhs) . Contain the component-wise forward and backward errors, respectively, for each solution vector.
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
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 hprfs interface are as follows:
ap
Holds the array A of size ( n*(n+1)/2 ).
afp
Holds the array AF of size ( n*(n+1)/2 ).
ipiv
Holds the vector of length n .
b
Holds the matrix B of size ( n,nrhs ).
x
Holds the matrix X of size ( n,nrhs ).
ferr
Holds the vector of length ( nrhs ).
berr
Holds the vector of length ( nrhs ).
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
Application Notes
The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.
For each right-hand side, computation of the backward error involves a minimum of \(16n^{2}\) operations. In addition, each step of iterative refinement involves \(24n^{2}\) operations; the number of iterations may range from 1 to 5.
Estimating the forward error involves solving a number of systems of linear equations \(A x = b\) ; the number is usually 4 or 5 and never more than 11. Each solution requires approximately \(8n^{2}\) floating-point operations.
The real counterpart of this routine is ?ssprfs (Refines the solution of a system of linear equations with a packed symmetric coefficient matrix and estimates the solution error.) / ?dsprfs (Refines the solution of a system of linear equations with a packed symmetric coefficient matrix and estimates the solution error.) .