Developer Reference for Intel® oneAPI Math Kernel Library for C
?tprfs
Estimates the error in the solution of a system of linear equations with a packed triangular coefficient matrix.
Syntax
lapack_int LAPACKE_stprfs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , const float*ap , const float*b , lapack_intldb , const float*x , lapack_intldx , float*ferr , float*berr );
lapack_int LAPACKE_dtprfs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , const double*ap , const double*b , lapack_intldb , const double*x , lapack_intldx , double*ferr , double*berr );
lapack_int LAPACKE_ctprfs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , const lapack_complex_float*ap , const lapack_complex_float*b , lapack_intldb , const lapack_complex_float*x , lapack_intldx , float*ferr , float*berr );
lapack_int LAPACKE_ztprfs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , const lapack_complex_double*ap , const lapack_complex_double*b , lapack_intldb , const lapack_complex_double*x , lapack_intldx , double*ferr , double*berr );
Include Files
mkl.h
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
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
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
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
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
The order of the matrix A ; n ≥ 0.
nrhs
The number of right-hand sides; nrhs ≥ 0.
ap , b , x , work
Arrays:
ap max(1, n(n+1)/2) contains the upper or lower triangular matrix A , as specified by uplo .
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 of size max(1, ldx * nrhs ) for column major layout and max(1, ldx * n ) for row major layout contains the solution matrix X .
ldb
The leading dimension of b ; ldb≥ max(1, n)ldb ≥ max(1, n ) for column major layout and ldb ≥ nrhs for row major layout .
ldx
The leading dimension of x ; ldx≥ max(1, n)ldb ≥ max(1, n ) for column major layout and ldb ≥ nrhs for row major layout .
iwork
Workspace array, size at least max(1, n) .
rwork
Workspace array, size at least max(1, n) .
Output Parameters
ferr , berr
Arrays, size at least max(1, nrhs) . Contain the component-wise forward and backward errors, respectively, for each solution vector.
Return Values
This function returns a value info .
If info = 0 , the execution is successful.
If info = -i , parameter i had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
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.