Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?pprfs
Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite coefficient matrix stored in a packed format and estimates its error.
Syntax
call spprfs ( uplo , n , nrhs , ap , afp , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dpprfs ( uplo , n , nrhs , ap , afp , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call cpprfs ( uplo , n , nrhs , ap , afp , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zpprfs ( uplo , n , nrhs , ap , afp , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call pprfs ( ap , afp , b , x [ , uplo ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric (Hermitian) positive definite 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
\(||x - x_{e}||_{\infty}/||x||_{\infty}\)
where x_{e} is the exact solution.
Before calling this routine:
call the factorization routine ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.)
call the solver routine ?pptrs (Solves a system of linear equations with a packed Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.) .
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates how the input matrix A has been factored:
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
REAL for spprfs
DOUBLE PRECISION for dpprfs
COMPLEX for cpprfs
DOUBLE COMPLEX for zpprfs .
Arrays:
ap(*) contains the original matrix A in a packed format, as supplied to ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.) .
afp(*) contains the factored matrix A in a packed format, as returned by ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.) .
b(ldb,*) contains the right-hand side matrix B .
x(ldx,*) 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, 3*n) for real flavors and max(1, 2*n) for complex flavors.
ap
ap contains the original matrix A in a packed format, as supplied to ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.) . The dimension of ap must be at least max(1,n(n+1)/2) .
afp
afp contains the factored matrix A in a packed format, as returned by ?pptrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.) . The dimension of afp must be at least max(1,n(n+1)/2) .
b
Array b of size max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout contains the right-hand side matrix B .
x
Array x of size max(1, ldx * nrhs ) for column major layout and max(1, ldx * n ) for row major layout contains the solution matrix X .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
ldx
INTEGER . The leading dimension of x ; ldx≥ max(1, n) .
iwork
INTEGER . Workspace array, size at least max(1, n) .
rwork
REAL for cpprfs
DOUBLE PRECISION for zpprfs .
Workspace array, size at least max(1, n) .
Output Parameters
x
The refined solution matrix X .
ferr , berr
REAL for single precision flavors.
DOUBLE PRECISION for double precision flavors.
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 pprfs 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).
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 \(4n^{2}\) floating-point operations (for real flavors) or \(16n^{2}\) operations (for complex flavors). In addition, each step of iterative refinement involves \(6n^{2}\) operations (for real flavors) or \(24n^{2}\) operations (for complex flavors); 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 of systems is usually 4 or 5 and never more than 11. Each solution requires approximately \(2n^{2}\) floating-point operations for real flavors or \(8n^{2}\) for complex flavors.