Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?syrfs
Refines the solution of a system of linear equations with a symmetric coefficient matrix and estimates its error.
Syntax
call ssyrfs ( uplo , n , nrhs , a , lda , af , ldaf , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dsyrfs ( uplo , n , nrhs , a , lda , af , ldaf , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call csyrfs ( uplo , n , nrhs , a , lda , af , ldaf , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zsyrfs ( uplo , n , nrhs , a , lda , af , ldaf , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call syrfs ( a , af , ipiv , b , x [ , uplo ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
ssyrfs dsyrfs csyrfs zsyrfs syrfs
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric full-storage 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.
a , af , b , x , work
REAL for ssyrfs
DOUBLE PRECISION for dsyrfs
COMPLEX for csyrfs
DOUBLE COMPLEX for zsyrfs .
Arrays:
a(lda,*) contains the original matrix A , as supplied to ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
af(ldaf,*) contains the factored matrix A , as returned by ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
b(ldb,*) contains the right-hand side matrix B .
x(ldx,*) contains the solution matrix X .
work(*) is a workspace array.
The second dimension of a and af must be at least max(1, n) ; 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.
a
Array a (size max(1, lda * n )) contains the original matrix A , as supplied to ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
af
Array af (size max(1, ldaf * n )) contains the factored matrix A , as returned by ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
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 .
lda
INTEGER . The leading dimension of a ; lda≥ max(1, n) .
ldaf
INTEGER . The leading dimension of af ; ldaf≥ max(1, 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 ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
iwork
INTEGER . Workspace array, size at least max(1, n) .
rwork
REAL for csyrfs
DOUBLE PRECISION for zsyrfs .
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 syrfs interface are as follows:
a
Holds the matrix A of size ( n,n ).
af
Holds the matrix AF of size ( n,n ).
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 \(4n^{2}\) 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 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.