Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?pbrfs
Refines the solution of a system of linear equations with a band symmetric (Hermitian) positive-definite coefficient matrix and estimates its error.
Syntax
call spbrfs ( uplo , n , kd , nrhs , ab , ldab , afb , ldafb , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dpbrfs ( uplo , n , kd , nrhs , ab , ldab , afb , ldafb , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call cpbrfs ( uplo , n , kd , nrhs , ab , ldab , afb , ldafb , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zpbrfs ( uplo , n , kd , nrhs , ab , ldab , afb , ldafb , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call pbrfs ( ab , afb , b , x [ , uplo ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
spbrfs dpbrfs cpbrfs zpbrfs pbrfs
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric (Hermitian) positive definite band 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’ .
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.
kd
INTEGER . The number of superdiagonals or subdiagonals in the matrix A ; kd ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ab , afb , b , x , work
REAL for spbrfs
DOUBLE PRECISION for dpbrfs
COMPLEX for cpbrfs
DOUBLE COMPLEX for zpbrfs .
Arrays:
ab(ldab,*) contains the original band matrix A , as supplied to ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.) .
afb(ldafb,*) contains the factored band matrix A , as returned by ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band 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 ab and afb 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.
ab
Array ab (size max( ldab * n )) contains the original band matrix A , as supplied to ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.) .
afb
Array afb (size max( ldafb * n )) contains the factored band matrix A , as returned by ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band 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 .
ldab
INTEGER . The leading dimension of ab ; ldab ≥ kd + 1.
ldafb
INTEGER . The leading dimension of afb ; ldafb ≥ kd + 1.
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 cpbrfs
DOUBLE PRECISION for zpbrfs .
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 pbrfs interface are as follows:
ab
Holds the array A of size ( kd +1, n ).
afb
Holds the array AF of size ( kd +1, 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 8n*kd floating-point operations (for real flavors) or 32n*kd operations (for complex flavors). In addition, each step of iterative refinement involves 12n*kd operations (for real flavors) or 48n*kd 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 4n*kd floating-point operations for real flavors or 16n*kd for complex flavors.