NAG CL Interface
f08ftc (zungtr)

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1 Purpose

f08ftc generates the complex unitary matrix Q, which was determined by f08fsc when reducing a Hermitian matrix to tridiagonal form.

2 Specification

#include <nag.h>
void  f08ftc (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Complex tau[], NagError *fail)
The function may be called by the names: f08ftc, nag_lapackeig_zungtr or nag_zungtr.

3 Description

f08ftc is intended to be used after a call to f08fsc, which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. f08fsc represents the unitary matrix Q as a product of n-1 elementary reflectors.
This function may be used to generate Q explicitly as a square matrix.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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: this must be the same argument uplo as supplied to f08fsc.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix Q.
Constraint: n0.
4: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: details of the vectors which define the elementary reflectors, as returned by f08fsc.
On exit: the n×n unitary matrix Q.
5: 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).
6: tau[dim] const Complex Input
Note: the dimension, dim, of the array tau must be at least max(1,n-1).
On entry: further details of the elementary reflectors, as returned by f08fsc.
7: 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, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
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.

7 Accuracy

The computed matrix Q differs from an exactly unitary matrix by a matrix E such that
E2 = O(ε) ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08ftc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ftc 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 163n3.
The real analogue of this function is f08ffc.

10 Example

This example computes all the eigenvalues and eigenvectors of the matrix A, where
A = ( -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ) .  
Here A is Hermitian and must first be reduced to tridiagonal form by f08fsc. The program then calls f08ftc to form Q, and passes this matrix to f08jsc which computes the eigenvalues and eigenvectors of A.

10.1 Program Text

Program Text (f08ftce.c)

10.2 Program Data

Program Data (f08ftce.d)

10.3 Program Results

Program Results (f08ftce.r)