NAG CL Interface
f07tsc (ztrtrs)

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1 Purpose

f07tsc solves a complex triangular system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B.

2 Specification

#include <nag.h>
void  f07tsc (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Integer nrhs, const Complex a[], Integer pda, Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07tsc, nag_lapacklin_ztrtrs or nag_ztrtrs.

3 Description

f07tsc solves a complex triangular system of linear equations AX=B, ATX=B or AHX=B.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies whether A is upper or lower triangular.
uplo=Nag_Upper
A is upper triangular.
uplo=Nag_Lower
A is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: trans Nag_TransType Input
On entry: indicates the form of the equations.
trans=Nag_NoTrans
The equations are of the form AX=B.
trans=Nag_Trans
The equations are of the form ATX=B.
trans=Nag_ConjTrans
The equations are of the form AHX=B.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4: diag Nag_DiagType Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag=Nag_NonUnitDiag
A is a nonunit triangular matrix.
diag=Nag_UnitDiag
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag=Nag_NonUnitDiag or Nag_UnitDiag.
5: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
6: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7: a[dim] const Complex Input
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: the n×n triangular matrix A.
If order=Nag_ColMajor, Aij is stored in a[(j-1)×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[(i-1)×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
If diag=Nag_UnitDiag, the diagonal elements of A are assumed to be 1, and are not referenced.
8: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax(1,n).
9: b[dim] Complex Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: the n×r solution matrix X.
10: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
11: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
Element value of the diagonal is exactly zero. A is singular and the solution has not been computed.

7 Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations (A+E)x=b, where
|E|c(n)ε|A| ,  
c(n) is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x c(n)cond(A,x)ε ,   provided   c(n)cond(A,x)ε<1 ,  
where cond(A,x)=|A-1||A||x|/x.
Note that cond(A,x)cond(A)=|A-1||A|κ(A); cond(A,x) can be much smaller than cond(A) and it is also possible for cond(AH), which is the same as cond(AT), to be much larger (or smaller) than cond(A).
Forward and backward error bounds can be computed by calling f07tvc, and an estimate for κ(A) can be obtained by calling f07tuc with norm=Nag_InfNorm.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07tsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately 4n2r.
The real analogue of this function is f07tec.

10 Example

This example solves the system of equations AX=B, where
A= ( 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i )  
and
B= ( -14.78-32.36i -18.02+28.46i 2.98-02.14i 14.22+15.42i -20.96+17.06i 5.62+35.89i 9.54+09.91i -16.46-01.73i ) .  

10.1 Program Text

Program Text (f07tsce.c)

10.2 Program Data

Program Data (f07tsce.d)

10.3 Program Results

Program Results (f07tsce.r)