Developer Reference for Intel® oneAPI Math Kernel Library for C
?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
lapack_int LAPACKE_spprfs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , const float*ap , const float*afp , const float*b , lapack_intldb , float*x , lapack_intldx , float*ferr , float*berr );
lapack_int LAPACKE_dpprfs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , const double*ap , const double*afp , const double*b , lapack_intldb , double*x , lapack_intldx , double*ferr , double*berr );
lapack_int LAPACKE_cpprfs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , const lapack_complex_float*ap , const lapack_complex_float*afp , const lapack_complex_float*b , lapack_intldb , lapack_complex_float*x , lapack_intldx , float*ferr , float*berr );
lapack_int LAPACKE_zpprfs ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , const lapack_complex_double*ap , const lapack_complex_double*afp , const lapack_complex_double*b , lapack_intldb , lapack_complex_double*x , lapack_intldx , double*ferr , double*berr );
Include Files
mkl.h
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
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
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
The order of the matrix A ; n ≥ 0.
nrhs
The number of right-hand sides; nrhs ≥ 0.
ap , afp , b , x , work
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
The leading dimension of b ; ldb≥ max(1, n)ldb ≥ max(1, n ) for column major layout and ldb ≥ nrhs for row major layout .
ldx
The leading dimension of x ; ldx≥ max(1, n)ldx ≥ max(1, n ) for column major layout and ldx ≥ nrhs for row major layout .
iwork
Workspace array, size at least max(1, n) .
rwork
Workspace array, size at least max(1, n) .
Output Parameters
x
The refined solution matrix X .
ferr , berr
Arrays, size at least max(1, nrhs) . Contain the component-wise forward and 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.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
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.