NAG Library Function Document

nag_zpttrs (f07jsc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zpttrs (f07jsc) computes the solution to a complex system of linear equations AX=B , where A  is an n  by n  Hermitian positive definite tridiagonal matrix and X  and B  are n  by r  matrices, using the LDLH  factorization returned by nag_zpttrf (f07jrc).

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zpttrs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, const double d[], const Complex e[], Complex b[], Integer pdb, NagError *fail)

3
Description

nag_zpttrs (f07jsc) should be preceded by a call to nag_zpttrf (f07jrc), which computes a modified Cholesky factorization of the matrix A  as
A=LDLH ,  
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix, with positive diagonal elements. nag_zpttrs (f07jsc) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UHDU , where U  is a unit upper bidiagonal matrix.

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:     uplo Nag_UploTypeInput
On entry: specifies the form of the factorization as follows:
uplo=Nag_Upper
A=UHDU.
uplo=Nag_Lower
A=LDLH.
Constraint: uplo=Nag_Upper or Nag_Lower.
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:     d[dim] const doubleInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLH or UHDU factorization of A.
6:     e[dim] const ComplexInput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: if uplo=Nag_Upper, e must contain the n-1 superdiagonal elements of the unit upper bidiagonal matrix U from the UHDU factorization of A.
If uplo=Nag_Lower, e must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L from the LDLH factorization of A.
7:     b[dim] ComplexInput/Output
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.
On exit: the n by r solution matrix X.
8:     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.
9:     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.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,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
E1 =OεA1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^ - x 1 x1 κA E1 A1 ,  
where κA = A-11 A1 , 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.
Following the use of this function nag_zptcon (f07juc) can be used to estimate the condition number of A  and nag_zptrfs (f07jvc) can be used to obtain approximate error bounds.

8
Parallelism and Performance

nag_zpttrs (f07jsc) 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  is proportional to nr .
The real analogue of this function is nag_dpttrs (f07jec).

10
Example

This example solves the equations
AX=B ,  
where A  is the Hermitian positive definite tridiagonal matrix
A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0  
and
B = 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i .  

10.1
Program Text

Program Text (f07jsce.c)

10.2
Program Data

Program Data (f07jsce.d)

10.3
Program Results

Program Results (f07jsce.r)

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