Developer Reference for Intel® oneAPI Math Kernel Library for C
?trtrs
Solves a system of linear equations with a triangular coefficient matrix, with multiple right-hand sides.
Syntax
lapack_intLAPACKE_strtrs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , constfloat*a , lapack_intlda , float*b , lapack_intldb );
lapack_intLAPACKE_dtrtrs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , constdouble*a , lapack_intlda , double*b , lapack_intldb );
lapack_intLAPACKE_ctrtrs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , constlapack_complex_float*a , lapack_intlda , lapack_complex_float*b , lapack_intldb );
lapack_intLAPACKE_ztrtrs ( intmatrix_layout , charuplo , chartrans , chardiag , lapack_intn , lapack_intnrhs , constlapack_complex_double*a , lapack_intlda , lapack_complex_double*b , lapack_intldb );
Include Files
mkl.h
Description
strtrs dtrtrs ctrtrs ztrtrs trtrs
The routine solves for X the following systems of linear equations with a triangular matrix A , with multiple right-hand sides stored in B :
A*X = B
if trans = ‘N’ ,
A^{T}*X = B
if trans = ‘T’ ,
A:code:`H`*X = B
if trans = ‘C’ (for complex matrices only).
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’ .
If trans = 'N' , then A*X = B is solved for X .
If trans = 'T' , then A:code:`T`*X = B is solved for X .
If trans = 'C' , then A:code:`H`*X = B is solved for X .
diag
Must be ‘N’ or ‘U’ .
If diag = 'N' , then A is not a unit triangular matrix.
If diag = 'U' , then A is unit triangular: diagonal elements of A are assumed to be 1 and not referenced in the array a .
n
The order of A ; the number of rows in B ; n ≥ 0.
nrhs
The number of right-hand sides; nrhs ≥ 0.
a
The array a(lda,*) contains the matrix A .
The size of a is max(1, lda * n ).
b
The array b(ldb,*) contains the matrix B whose columns are the right-hand sides for the systems of equations.
The size of b is max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout.
lda
The leading dimension of a ; lda≥ max(1, n) .
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 .
Output Parameters
b
Overwritten by the solution matrix X .
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
For each right-hand side \(b\) , the computed solution is the exact solution of a perturbed system of equations \((A + E)x = b\) , where
\[|E| \leq c(n) \varepsilon |A|\]
\(c(n)\) is a modest linear function of \(k\) , and \(\varepsilon\) is the machine precision.
If \(x_{0}\) is the true solution, the computed solution \(x\) satisfies this error bound:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(n) \operatorname{cond}(A, x) \varepsilon, \text{ provided } c(n) \operatorname{cond}(A,x)\varepsilon < 1\]
where
\[\operatorname{cond}(A,x) = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
Note that \(\operatorname{cond}(A,x)\) can be much smaller than \(\kappa_{\infty}(A)\) ; the condition number of \(A^{T}\) and \(A^{H}\) might or might not be equal to \(\kappa_{\infty}(A)\) .
The approximate number of floating-point operations for one right-hand side vector \(b\) is \(n^2\) for real flavors and \(4n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_{infty}(A)\) , call ?trcon (Estimates the reciprocal of the condition number of a triangular matrix.) .
To estimate the error in the solution, call ?trrfs (Estimates the error in the solution of a system of linear equations with a triangular coefficient matrix.) .