Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?gbrfs
Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.
Syntax
call sgbrfs ( trans , n , kl , ku , nrhs , ab , ldab , afb , ldafb , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dgbrfs ( trans , n , kl , ku , nrhs , ab , ldab , afb , ldafb , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call cgbrfs ( trans , n , kl , ku , nrhs , ab , ldab , afb , ldafb , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zgbrfs ( trans , n , kl , ku , nrhs , ab , ldab , afb , ldafb , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call gbrfs ( ab , afb , ipiv , b , x [ , kl ] [ , trans ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sgbrfs dgbrfs cgbrfs zgbrfs gbrfs
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B or A^{T}*X = B or A:code:`H`*X = B with a 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
trans
CHARACTER*1 . Must be ‘N’ or ‘T’ or ‘C’ .
Indicates the form of the equations:
If trans = 'N' , the system has the form A*X = B .
If trans = 'T' , the system has the form A:code:`T`*X = B .
If trans = 'C' , the system has the form A:code:`H`*X = B .
n
INTEGER . The order of the matrix A ; n ≥ 0.
kl
INTEGER . The number of sub-diagonals within the band of A ; kl ≥ 0.
ku
INTEGER . The number of super-diagonals within the band of A ; ku ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ab , afb , b , x , work
REAL for sgbrfs
DOUBLE PRECISION for dgbrfs
COMPLEX for cgbrfs
DOUBLE COMPLEX for zgbrfs .
Arrays:
ab (size ldab by *) contains the original band matrix A , as supplied to GUID-03E2AA41-0886-485D-B0D7-1C2186119220.xml#GUID-03E2AA41-0886-485D-B0D7-1C2186119220 , but stored in rows from 1 to kl + ku + 1 .
afb (size ldafb by *) contains the factored band matrix A , as returned by GUID-03E2AA41-0886-485D-B0D7-1C2186119220.xml#GUID-03E2AA41-0886-485D-B0D7-1C2186119220 .
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 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 .
ldab
INTEGER . The leading dimension of ab .
ldafb
INTEGER . The leading dimension of afb .
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 ?gbtrf (Computes the LU factorization of a general m-by-n band matrix.) .
iwork
INTEGER . Workspace array, size at least max(1, n) .
rwork
REAL for cgbrfs
DOUBLE PRECISION for zgbrfs.
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 gbrfs interface are as follows:
ab
Holds the array A of size ( kl+ku+1,n ).
afb
Holds the array AF of size ( 2*kl*ku+1,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 ).
trans
Must be ‘N’ , ‘C’ , or ‘T’ . The default value is ‘N’ .
kl
If omitted, assumed kl = ku .
ku
Restored as ku = lda-kl-1 .
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 4 n ( kl + ku ) floating-point operations (for real flavors) or 16 n ( kl + ku ) operations (for complex flavors). In addition, each step of iterative refinement involves 2 n (4 kl + 3 ku ) operations (for real flavors) or 8 n (4 kl + 3 ku ) 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.