NAG Library Function Document

nag_zgtrfs (f07cvc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zgtrfs (f07cvc) computes error bounds and refines the solution to a complex system of linear equations AX=B  or ATX=B  or AHX=B , where A  is an n  by n  tridiagonal matrix and X  and B  are n  by r  matrices, using the LU  factorization returned by nag_zgttrf (f07crc) and an initial solution returned by nag_zgttrs (f07csc). Iterative refinement is used to reduce the backward error as much as possible.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgtrfs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex dlf[], const Complex df[], const Complex duf[], const Complex du2[], const Integer ipiv[], const Complex b[], Integer pdb, Complex x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3
Description

nag_zgtrfs (f07cvc) should normally be preceded by calls to nag_zgttrf (f07crc) and nag_zgttrs (f07csc). nag_zgttrf (f07crc) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A  as
A=PLU ,  
where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element in each column, and U  is an upper triangular band matrix, with two superdiagonals. nag_zgttrs (f07csc) then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , nag_zgtrfs (f07cvc) computes a component-wise backward error, β , the smallest relative perturbation in each element of A  and b  such that x^  is the exact solution of a perturbed system
A+E x^=b+f , with  eij β aij , and  fj β bj .  
The function also estimates a bound for the component-wise forward error in the computed solution defined by max xi - xi^ / max xi^ , where x  is the corresponding column of the exact solution, X .

4
References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

5
Arguments

1:     order Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     trans Nag_TransTypeInput
On entry: specifies the equations to be solved as follows:
trans=Nag_NoTrans
Solve AX=B for X.
trans=Nag_Trans
Solve ATX=B for X.
trans=Nag_ConjTrans
Solve AHX=B for X.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     nrhs IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     dl[dim] const ComplexInput
Note: the dimension, dim, of the array dl must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
6:     d[dim] const ComplexInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the matrix A.
7:     du[dim] const ComplexInput
Note: the dimension, dim, of the array du must be at least max1,n-1.
On entry: must contain the n-1 superdiagonal elements of the matrix A.
8:     dlf[dim] const ComplexInput
Note: the dimension, dim, of the array dlf must be at least max1,n-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
9:     df[dim] const ComplexInput
Note: the dimension, dim, of the array df must be at least max1,n.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
10:   duf[dim] const ComplexInput
Note: the dimension, dim, of the array duf must be at least max1,n-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
11:   du2[dim] const ComplexInput
Note: the dimension, dim, of the array du2 must be at least max1,n-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
12:   ipiv[dim] const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipiv[i-1], and ipiv[i-1] must always be either i or i+1, ipiv[i-1]=i indicating that a row interchange was not performed.
13:   b[dim] const ComplexInput
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by r matrix of right-hand sides B.
14:   pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
15:   x[dim] ComplexInput/Output
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On entry: the n by r initial solution matrix X.
On exit: the n by r refined solution matrix X.
16:   pdx IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
17:   ferr[nrhs] doubleOutput
On exit: estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferr[j-1], where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is almost always a slight overestimate of the true error.
18:   berr[nrhs] doubleOutput
On exit: estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
19:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E=OεA  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x x κA E A ,  
where κA=A-1 A , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function nag_zgtcon (f07cuc) can be used to estimate the condition number of A .

8
Parallelism and Performance

nag_zgtrfs (f07cvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgtrfs (f07cvc) 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 floating-point operations required to solve the equations AX=B  or ATX=B  or AHX=B  is proportional to nr . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The real analogue of this function is nag_dgtrfs (f07chc).

10
Example

This example solves the equations
AX=B ,  
where A  is the tridiagonal matrix
A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i  
and
B = 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .  
Estimates for the backward errors and forward errors are also output.

10.1
Program Text

Program Text (f07cvce.c)

10.2
Program Data

Program Data (f07cvce.d)

10.3
Program Results

Program Results (f07cvce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017