Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?ptrfs
Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix and estimates its error.
Syntax
call sptrfs ( n , nrhs , d , e , df , ef , b , ldb , x , ldx , ferr , berr , work , info )
call dptrfs ( n , nrhs , d , e , df , ef , b , ldb , x , ldx , ferr , berr , work , info )
call cptrfs ( uplo , n , nrhs , d , e , df , ef , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call zptrfs ( uplo , n , nrhs , d , e , df , ef , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call ptrfs ( d , df , e , ef , b , x [ , ferr ] [ , berr ] [ , info ] )
call ptrfs ( d , df , e , ef , b , x [ , uplo ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sptrfs dptrfs cptrfs zptrfs ptrfs
The routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a symmetric (Hermitian) positive definite 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:
\(|\delta a_{ij}| \leq \beta|a_{ij}|, |\delta b_{i}| \leq \beta|b_{i}|\) 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:
call the factorization routine ?pttrf (Computes the factorization of a symmetric (Hermitian) positive-definite tridiagonal matrix.)
call the solver routine ?pttrs (Solves a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix using the factorization computed by ?pttrf.) .
Input Parameters
uplo
CHARACTER*1 . Used for complex flavors only. Must be ‘U’ or ‘L’ .
Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and how A is factored:
If uplo = 'U' , the array e stores the superdiagonal of A , and A is factored as U^{H}*D*U .
If uplo = 'L' , the array e stores the subdiagonal of A , and A is factored as L*D*L^{H} .
n
INTEGER . The order of the matrix A ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
d , df , rwork
REAL for single precision flavors DOUBLE PRECISION for double precision flavors
Arrays: d(n) , df(n) , rwork(n) .
The array d contains the n diagonal elements of the tridiagonal matrix A .
The array df contains the n diagonal elements of the diagonal matrix D from the factorization of A as computed by ?pttrf (Computes the factorization of a symmetric (Hermitian) positive-definite tridiagonal matrix.) .
The array rwork is a workspace array used for complex flavors only.
d
The array d (size n ) contains the n diagonal elements of the tridiagonal matrix A .
df
The array df (size n ) contains the n diagonal elements of the diagonal matrix D from the factorization of A as computed by ?pttrf (Computes the factorization of a symmetric (Hermitian) positive-definite tridiagonal matrix.) .
e , ef , b , x , work
REAL for sptrfs
DOUBLE PRECISION for dptrfs
COMPLEX for cptrfs
DOUBLE COMPLEX for zptrfs .
Arrays: e(n -1) , ef(n -1) , b(ldb,nrhs) , x(ldx,nrhs) , work(*) .
The array e contains the ( n - 1) off-diagonal elements of the tridiagonal matrix A (see uplo ).
The array ef contains the ( n - 1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ?pttrf (Computes the factorization of a symmetric (Hermitian) positive-definite tridiagonal matrix.) (see uplo ).
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
The array x contains the solution matrix X as computed by ?pttrs (Solves a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix using the factorization computed by ?pttrf.) .
The array work is a workspace array. The dimension of work must be at least 2*n for real flavors, and at least n for complex flavors.
e , ef , b , x
The array e (size n -1) contains the ( n - 1) off-diagonal elements of the tridiagonal matrix A (see uplo ).
The array ef (size n -1) contains the ( n - 1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ?pttrf (Computes the factorization of a symmetric (Hermitian) positive-definite tridiagonal matrix.) (see uplo ).
The array b of size max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout contains the matrix B whose columns are the right-hand sides for the systems of equations.
The 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 as computed by ?pttrs (Solves a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix using the factorization computed by ?pttrf.) .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
ldx
INTEGER . The leading dimension of x ; ldx≥ 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 ptrfs interface are as follows:
d
Holds the vector of length n .
df
Holds the vector of length n .
e
Holds the vector of length ( n -1).
ef
Holds the vector of length ( n -1).
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
Used in complex flavors only. Must be ‘U’ or ‘L’ . The default value is ‘U’ .