NAG Library Function Document

nag_ztptri (f07uwc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_ztptri (f07uwc) computes the inverse of a complex triangular matrix, using packed storage.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_ztptri (Nag_OrderType order, Nag_UploType uplo, Nag_DiagType diag, Integer n, Complex ap[], NagError *fail)

3
Description

nag_ztptri (f07uwc) forms the inverse of a complex triangular matrix A, using packed storage. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

4
References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

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 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:     diag Nag_DiagTypeInput
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.
4:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     ap[dim] ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n triangular matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
If diag=Nag_UnitDiag, the diagonal elements of AP are assumed to be 1, and are not referenced; the same storage scheme is used whether diag=Nag_NonUnitDiag or diag=Nag_UnitDiag.
On exit: A is overwritten by A-1, using the same storage format as described above.
6:     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.
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.
NE_SINGULAR
Element value of the diagonal is exactly zero. A is singular its inverse cannot be computed.

7
Accuracy

The computed inverse X satisfies
XA-IcnεXA ,  
where cn is a modest linear function of n, and ε is the machine precision.
Note that a similar bound for AX-I cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
X-A-1cnεA-1AX .  
See Du Croz and Higham (1992).

8
Parallelism and Performance

nag_ztptri (f07uwc) 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 43n3.
The real analogue of this function is nag_dtptri (f07ujc).

10
Example

This example computes the inverse of the matrix A, 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 ,  
using packed storage.

10.1
Program Text

Program Text (f07uwce.c)

10.2
Program Data

Program Data (f07uwce.d)

10.3
Program Results

Program Results (f07uwce.r)

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