Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?tprfs
Estimates the error in the solution of a system of linear equations with a packed triangular coefficient matrix.
Syntax
call stprfs ( uplo , trans , diag , n , nrhs , ap , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call dtprfs ( uplo , trans , diag , n , nrhs , ap , b , ldb , x , ldx , ferr , berr , work , iwork , info )
call ctprfs ( uplo , trans , diag , n , nrhs , ap , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call ztprfs ( uplo , trans , diag , n , nrhs , ap , b , ldb , x , ldx , ferr , berr , work , rwork , info )
call tprfs ( ap , b , x [ , uplo ] [ , trans ] [ , diag ] [ , ferr ] [ , berr ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
stprfs dtprfs ctprfs ztprfs tprfs
The routine estimates the errors in the solution to a system of linear equations A*X = B or A^{T}*X = B or A:code:`H`*X = B with a packed triangular 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 \((A + \delta A)x = (b + \delta b)\) .
The routine also 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 solver routine ?tptrs (Solves a system of linear equations with a packed triangular coefficient matrix, with multiple right-hand sides.) .
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates whether A is upper or lower triangular:
If uplo = 'U' , then A is upper triangular.
If uplo = 'L' , then A is lower triangular.
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 .
diag
CHARACTER*1 . Must be ‘N’ or ‘U’ .
If diag = 'N' , A is not a unit triangular matrix.
If diag = 'U' , A is unit triangular: diagonal elements of A are assumed to be 1 and not referenced in the array ap .
n
INTEGER . The order of the matrix A ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ap , b , x , work
REAL for stprfs
DOUBLE PRECISION for dtprfs
COMPLEX for ctprfs
DOUBLE COMPLEX for ztprfs .
Arrays:
ap (size *) contains the upper or lower triangular matrix A , as specified by uplo .
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 dimension of ap must be at least max(1,n(n+1)/2) ; 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 .
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 ctprfs
DOUBLE PRECISION for ztprfs .
Workspace array, size at least max(1, n) .
Output Parameters
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 tprfs interface are as follows:
ap
Holds the array A of size ( n*(n+1)/2 ).
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’ .
trans
Must be ‘N’ , ‘C’ , or ‘T’ . The default value is ‘N’ .
diag
Must be ‘N’ or ‘U’ . The default value is ‘N’ .
Application Notes
The bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual error.
A call to this routine involves, for each right-hand side, solving a number of systems of linear equations \(A x = b\) ; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately \(n^{2}\) floating-point operations for real flavors or \(4n^{2}\) for complex flavors.