NAG Library Function Document

nag_dorgtr (f08ffc)

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_dsytrd (f08fec).
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[ $\dim$ ] – doubleInput/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_dsytrd (f08fec).

On exit: the $n$ by $n$ orthogonal 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 doubleInput: 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_dsytrd (f08fec).
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 orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8 Parallelism and Performance

nag_dorgtr (f08ffc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dorgtr (f08ffc) 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 floating-point operations is approximately

\frac{4}{3} n^{3}

The complex analogue of this function is nag_zungtr (f08ftc).

10 Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{array}) .

Here

A

is symmetric and must first be reduced to tridiagonal form by nag_dsytrd (f08fec). The program then calls nag_dorgtr (f08ffc) to form

Q

, and passes this matrix to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of

A

NAG Library Function Documentnag_dorgtr (f08ffc)

+− 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_dorgtr (f08ffc)