NAG Library Function Document

nag_zungtr (f08ftc)

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $order = Nag_RowMajor$ or $Nag_ColMajor$ .
2: uplo – Nag_UploTypeInput: On entry: this must be the same argument uplo as supplied to nag_zhetrd (f08fsc).
Constraint: $uplo = Nag_Upper$ or $Nag_Lower$ .
3: n – IntegerInput: On entry: $n$ , the order of the matrix $Q$ .
Constraint: $n \geq 0$ .
4: a[ $\dim$ ] – ComplexInput/Output: Note: the dimension, dim, of the array a must be at least $\max (1, pda \times n)$ .
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zhetrd (f08fsc).

On exit: the $n$ by $n$ unitary matrix $Q$ .
If $order = Nag_ColMajor$ , the $(i, j)$ th element of the matrix is stored in $a [(j - 1) \times pda + i - 1]$ .
If $order = Nag_RowMajor$ , the $(i, j)$ th element of the matrix is stored in $a [(i - 1) \times pda + j - 1]$ .
5: pda – IntegerInput: On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.

Constraint: $pda \geq \max (1, n)$ .
6: tau[ $\dim$ ] – const ComplexInput: 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 nag_zhetrd (f08fsc).
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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.

7 Accuracy

The computed matrix

Q

differs from an exactly unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Parallelism and Performance

nag_zungtr (f08ftc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zungtr (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately

\frac{16}{3} n^{3}

The real analogue of this function is nag_dorgtr (f08ffc).

10 Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} - 2.28 + 0.00 i & 1.78 - 2.03 i & 2.26 + 0.10 i & - 0.12 + 2.53 i \\ 1.78 + 2.03 i & - 1.12 + 0.00 i & 0.01 + 0.43 i & - 1.07 + 0.86 i \\ 2.26 - 0.10 i & 0.01 - 0.43 i & - 0.37 + 0.00 i & 2.31 - 0.92 i \\ - 0.12 - 2.53 i & - 1.07 - 0.86 i & 2.31 + 0.92 i & - 0.73 + 0.00 i \end{array}) .

Here

A

is Hermitian and must first be reduced to tridiagonal form by nag_zhetrd (f08fsc). The program then calls nag_zungtr (f08ftc) to form

Q

, and passes this matrix to nag_zsteqr (f08jsc) which computes the eigenvalues and eigenvectors of

A

NAG Library Function Documentnag_zungtr (f08ftc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_zungtr (f08ftc)