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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dormtr (f08fg)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dormtr (f08fg) multiplies an arbitrary real matrix

C

by the real orthogonal matrix

Q

which was determined by nag_lapack_dsytrd (f08fe) when reducing a real symmetric matrix to tridiagonal form.

Syntax

[c, info] = f08fg(side, uplo, trans, a, tau, c, 'm', m, 'n', n)

[c, info] = nag_lapack_dormtr(side, uplo, trans, a, tau, c, 'm', m, 'n', n)

Description

nag_lapack_dormtr (f08fg) is intended to be used after a call to nag_lapack_dsytrd (f08fe), which reduces a real symmetric matrix

A

to symmetric tridiagonal form

T

by an orthogonal similarity transformation:

A = Q T Q^{T}

. nag_lapack_dsytrd (f08fe) represents the orthogonal matrix

Q

as a product of elementary reflectors.

This function may be used to form one of the matrix products

Q C, Q^{T} C, C Q ​ or ​ C Q^{T},

overwriting the result on

C

(which may be any real rectangular matrix).

A common application of this function is to transform a matrix

Z

of eigenvectors of

T

to the matrix

QZ

of eigenvectors of

A

References

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

Parameters

Compulsory Input Parameters

1: $side$ – string (length ≥ 1)

Indicates how

Q

Q^{T}

is to be applied to

C

$side ='L'$: $Q$ or $Q^{T}$ is applied to $C$ from the left.
$side ='R'$: $Q$ or $Q^{T}$ is applied to $C$ from the right.

Constraint:

side ='L'

'R'

2: $uplo$ – string (length ≥ 1)

This must be the same argument uplo as supplied to nag_lapack_dsytrd (f08fe).

Constraint:

uplo ='U'

'L'

3: $trans$ – string (length ≥ 1)

Indicates whether

Q

Q^{T}

is to be applied to

C

$trans ='N'$: $Q$ is applied to $C$ .
$trans ='T'$: $Q^{T}$ is applied to $C$ .

Constraint:

trans ='N'

'T'

4: $a (lda :)$ – double array

The first dimension,

lda

, of the array a must satisfy

if $side ='L'$ , $lda \geq \max (1, m)$ ;
if $side ='R'$ , $lda \geq \max (1, n)$ .

The second dimension of the array a must be at least

\max (1, m)

side ='L'

and at least

\max (1, n)

side ='R'

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsytrd (f08fe).

5: $tau (:)$ – double array

The dimension of the array tau must be at least

\max (1, m - 1)

side ='L'

and at least

\max (1, n - 1)

side ='R'

Further details of the elementary reflectors, as returned by nag_lapack_dsytrd (f08fe).

6: $c (ldc :)$ – double array

The first dimension of the array c must be at least

\max (1, m)

The second dimension of the array c must be at least

\max (1, n)

The

m

n

matrix

C

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array c.
$m$ , the number of rows of the matrix $C$ ; $m$ is also the order of $Q$ if $side ='L'$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array c.
$n$ , the number of columns of the matrix $C$ ; $n$ is also the order of $Q$ if $side ='R'$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $c (ldc :)$ – double array: The first dimension of the array c will be $\max (1, m)$ .
The second dimension of the array c will be $\max (1, n)$ .

c stores $Q C$ or $Q^{T} C$ or $C Q$ or $C Q^{T}$ as specified by side and trans.
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: side, 2: uplo, 3: trans, 4: m, 5: n, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

2 m^{2} n

side ='L'

and

2 m n^{2}

side ='R'

The complex analogue of this function is nag_lapack_zunmtr (f08fu).

Example

This example computes the two smallest eigenvalues, and the associated 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

T

by nag_lapack_dsytrd (f08fe). The program then calls nag_lapack_dstebz (f08jj) to compute the requested eigenvalues and nag_lapack_dstein (f08jk) to compute the associated eigenvectors of

T

. Finally nag_lapack_dormtr (f08fg) is called to transform the eigenvectors to those of

A

Open in the MATLAB editor: f08fg_example

function f08fg_example


fprintf('f08fg example results\n\n');

% Find eigenvalues 1:2 of A
a = [ 2.07,  0,    0,     0;
      3.87, -0.21, 0,     0;
      4.20,  1.87, 1.15,  0;
     -1.15,  0.63, 2.06, -1.81];

% A --> Q T Q^T, for tridiagonal T with diagonal Td and off-diagonal Tod.
uplo = 'L';
[QT, Td, Tod, tau, info] = f08fe( ...
				  uplo, a);

% Get eigenvalues 1:2 of T (= eigenvalues of A)
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, ~, w, iblock, isplit, info] = ...
  f08jj(...
        'I', 'B', vl, vu, il, iu, abstol, Td, Tod);

% Get corresponding eigenvectors of T
[v, ifailv, info] = f08jk( ...
			   Td, Tod, m, w, iblock, isplit);

% Transform Q*V to get eigenvectors of A 
side = 'Left';
trans = 'No transpose';
[z, info] = f08fg( ...
                   side, uplo, trans, QT, tau, v);

fprintf(' Eigenvalues numbered 1 to 2 are:\n   ');
fprintf(' %7.4f',w(1:m));
fprintf('\n\n');

[ifail] = x04ca( ...
		 'General', ' ', z, 'Corresponding eigenvectors of A');

f08fg example results

 Eigenvalues numbered 1 to 2 are:
    -5.0034 -1.9987

 Corresponding eigenvectors of A
          1       2
 1   0.5658 -0.2328
 2  -0.3478  0.7994
 3  -0.4740 -0.4087
 4   0.5781  0.3737

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox