Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?gtrfs
Refines the solution of a system of linear equations with a tridiagonal coefficient matrix and estimates its error.
Syntax
call sgtrfs ( trans , n , nrhs , dl , d , du , dlf , df , duf , du2 , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dgtrfs ( trans , n , nrhs , dl , d , du , dlf , df , duf , du2 , ipiv , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call cgtrfs ( trans , n , nrhs , dl , d , du , dlf , df , duf , du2 , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zgtrfs ( trans , n , nrhs , dl , d , du , dlf , df , duf , du2 , ipiv , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call gtrfs ( dl , d , du , dlf , df , duf , du2 , ipiv , b , x [ , trans ] [ , 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 or A^{T}*X = B or A:code:`H`*X = B with a tridiagonal 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:
|δa_{ij}|/|a_{ij}| ≤β|a_{ij}|, |δb_{i}|/|b_{i}| ≤β|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.
nrhs
INTEGER . The number of right-hand sides, that is, the number of columns of the matrix B ; nrhs ≥ 0.
dl , d , du , dlf , df , duf , du2 , b , x , work
REAL for sgtrfs
DOUBLE PRECISION for dgtrfs
COMPLEX for cgtrfs
DOUBLE COMPLEX for zgtrfs .
Arrays:
dl , dimension ( n -1), contains the subdiagonal elements of A .
d , dimension ( n ), contains the diagonal elements of A .
du , dimension ( n -1), contains the superdiagonal elements of A .
dlf , dimension ( n -1), contains the ( n - 1) multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
df , dimension ( n ), contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A .
duf , dimension ( n -1), contains the ( n - 1) elements of the first superdiagonal of U .
du2 , dimension ( n -2), contains the ( n - 2) elements of the second superdiagonal of U .
b(ldb,nrhs) contains the right-hand side matrix B .
x(ldx,nrhs) contains the solution matrix X , as computed by ?gttrs (Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by ?gttrf.) .
work(*) is a workspace array; the dimension of work must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors.
dl
Array dl of size n -1 contains the subdiagonal elements of A .
d
Array d of size n contains the diagonal elements of A .
du
Array du of size n -1 contains the superdiagonal elements of A .
dlf
Array dlf of size n -1 contains the ( n - 1) multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
df
Array df of size n contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A .
duf
Array duf of size n -1 contains the ( n - 1) elements of the first superdiagonal of U .
du2
Array du2 of size n -2 contains the ( n - 2) elements of the second superdiagonal of U .
b
Array b (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 (size max(1, ldx * nrhs ) for column major layout and max(1, ldx * n ) contains the solution matrix X , as computed by ?gttrs (Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by ?gttrf.) .
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 ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
iwork
INTEGER . Workspace array, size ( n ). Used for real flavors only.
rwork
REAL for cgtrfs
DOUBLE PRECISION for zgtrfs .
Workspace array, size ( n ). Used for complex flavors only.
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 gtrfs interface are as follows:
dl
Holds the vector of length ( n -1).
d
Holds the vector of length n .
du
Holds the vector of length ( n -1).
dlf
Holds the vector of length ( n -1).
df
Holds the vector of length n .
duf
Holds the vector of length ( n -1).
du2
Holds the vector of length ( n -2).
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’ .