Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?trtrs
Solves a system of linear equations with a triangular coefficient matrix, with multiple right-hand sides.
Syntax
call strtrs ( uplo , trans , diag , n , nrhs , a , lda , b , ldb , info )
call dtrtrs ( uplo , trans , diag , n , nrhs , a , lda , b , ldb , info )
call ctrtrs ( uplo , trans , diag , n , nrhs , a , lda , b , ldb , info )
call ztrtrs ( uplo , trans , diag , n , nrhs , a , lda , b , ldb , info )
call trtrs ( a , b [ , uplo ] [ , trans ] [ , diag ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
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
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’ .
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
CHARACTER*1 . 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
INTEGER . The order of A ; the number of rows in B ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
a
REAL for strtrs
DOUBLE PRECISION for dtrtrs
COMPLEX for ctrtrs
DOUBLE COMPLEX for ztrtrs .
The array a(lda,*) contains the matrix A .
The second dimension of a must be at least max(1,n) .
b
REAL for strtrs
DOUBLE PRECISION for dtrtrs
COMPLEX for ctrtrs
DOUBLE COMPLEX for ztrtrs .
The array b(ldb,*) contains the matrix B whose columns are the right-hand sides for the systems of equations.
The second dimension of b at least max(1,nrhs) .
lda
INTEGER . The leading dimension of a ; lda≥ max(1, n) .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
Output Parameters
b
Overwritten by the solution matrix X .
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 trtrs interface are as follows:
a
Stands for argument ap in FORTRAN 77 interface. Holds the matrix A of size (n*(n+1)/2) .
b
Holds the matrix B of size ( n , 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
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.) .