. A nonsingular matrix has multiple square roots. For a matrix with no eigenvalues on the closed negative real line, the principal square root, denoted by

A^{1 / 2}

, is the unique square root whose eigenvalues lie in the open right half-plane.

nag_matop_real_tri_matrix_sqrt (f01ep) computes

A^{1 / 2}

, where

A

is an upper quasi-triangular matrix, with

1 \times 1

and

2 \times 2

blocks on the diagonal. Such matrices arise from the Schur factorization of a real general matrix, as computed by nag_lapack_dhseqr (f08pe), for example. nag_matop_real_tri_matrix_sqrt (f01ep) does not require

A

to be in the canonical Schur form described in nag_lapack_dhseqr (f08pe), it merely requires

A

to be upper quasi-triangular.

A^{1 / 2}

then has the same block triangular structure as

A

The algorithm used by nag_matop_real_tri_matrix_sqrt (f01ep) is described in Higham (1987). In addition a blocking scheme described in Deadman et al. (2013) is used.

References

Björck Å and Hammarling S (1983) A Schur method for the square root of a matrix Linear Algebra Appl. 52/53 127–140

Deadman E, Higham N J and Ralha R (2013) Blocked Schur Algorithms for Computing the Matrix Square Root Applied Parallel and Scientific Computing: 11th International Conference, (PARA 2012, Helsinki, Finland) P. Manninen and P. Öster, Eds Lecture Notes in Computer Science 7782 171–181 Springer–Verlag

Higham N J (1987) Computing real square roots of a real matrix Linear Algebra Appl. 88/89 405–430

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $n$ .
The second dimension of the array a must be at least $n$ .

The $n$ by $n$ upper quasi-triangular matrix $A$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $n$ .
The second dimension of the array a will be $n$ .

The $n$ by $n$ principal matrix square root $A^{1 / 2}$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: $A$ has negative or vanishing eigenvalues. The principal square root is not defined in this case. nag_matop_real_gen_matrix_sqrt (f01en) or nag_matop_complex_gen_matrix_sqrt (f01fn) may be able to provide further information.

$ifail = 2$: An internal error occurred. It is likely that the function was called incorrectly.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 3$: Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computed square root

\hat{X}

satisfies

{\hat{X}}^{2} = A + Δ A

, where

{‖Δ A‖}_{F} \approx O (ε) n {‖\hat{X}‖}_{F}^{2}

, where

ε

is machine precision.

Further Comments

The cost of the algorithm is

n^{3} / 3

floating-point operations; see Algorithm 6.7 of Higham (2008).

O (n)

of integer allocatable memory is required by the function.

A

is a full matrix, then nag_matop_real_gen_matrix_sqrt (f01en) should be used to compute the square root. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_sqrt (f01fn) can be used to return a complex, non-principal square root.

If condition number and residual bound estimates are required, then nag_matop_real_gen_matrix_cond_sqrt (f01jd) should be used. For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008).

Example

This example finds the principal matrix square root of the matrix

A = (\begin{array}{r} 6 & 4 & - 5 & 15 \\ 8 & 6 & - 3 & 10 \\ 0 & 0 & 3 & - 4 \\ 0 & 0 & 4 & 3 \end{array}) .

Open in the MATLAB editor: f01ep_example

function f01ep_example


fprintf('f01ep example results\n\n');

% Principal square root of matrix A

a = [ 6  4 -5 15;
      8  6 -3 10;
      0  0  3 -4;
      0  0  4  3];

[as, ifail] = f01ep(a);

disp('Square root of A:');
disp(as);

f01ep example results

Square root of A:
    2.0000    1.0000   -2.0000    3.0000
    2.0000    2.0000    0.0000    1.0000
         0         0    2.0000   -1.0000
         0         0    1.0000    2.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox