. 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.

A^{1 / 2}

is computed using the algorithm described in Higham (1987). This is a real arithmetic version of the algorithm of Björck and Hammarling (1983). 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 $n$ 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 $n$ .

Contains, if $ifail = 0$ , the $n$ by $n$ principal matrix square root, $A^{1 / 2}$ . Alternatively, if $ifail = 1$ , contains an $n$ by $n$ non-principal square root of $A$ .
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 a semisimple vanishing eigenvalue. A non-principal square root is returned.

$ifail = 2$: $A$ has a defective vanishing eigenvalue. The square root cannot be found in this case.

$ifail = 3$: $A$ has a negative real eigenvalue. The principal square root is not defined. nag_matop_complex_gen_matrix_sqrt (f01fn) can be used to return a complex, non-principal square root.

$ifail = 4$: 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^{3} {‖\hat{X}‖}_{F}^{2}

, where

ε

is machine precision.

For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008).

Further Comments

The cost of the algorithm is

85 n^{3} / 3

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

O (2 \times n^{2})

of real allocatable memory is required by the function.

If condition number and residual bound estimates are required, then nag_matop_real_gen_matrix_cond_sqrt (f01jd) should be used.

Example

This example finds the principal matrix square root of the matrix

A = (\begin{array}{r} 507 & 622 & 300 & - 202 \\ 237 & 352 & 126 & - 60 \\ 751 & 950 & 440 & - 286 \\ - 286 & - 326 & - 192 & 150 \end{array}) .

Open in the MATLAB editor: f01en_example

function f01en_example


fprintf('f01en example results\n\n');

% Principal square root of matrix A

a = [ 507  622  300 -202;
      237  352  126  -60;
      751  950  440 -286;
     -286 -326 -192  150];

[as, ifail] = f01en(a);

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

f01en example results

Square root of A:
   15.0000   14.0000    8.0000   -6.0000
    6.0000   14.0000    3.0000    0.0000
   21.0000   24.0000   12.0000   -8.0000
   -5.0000   -4.0000   -7.0000    8.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox