. 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 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 (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – complex 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: $a (lda :)$ – complex 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 negative or 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.

$ifail = 3$: 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.

Further Comments

The cost of the algorithm is

85 n^{3} / 3

complex floating-point operations; see Algorithm 6.3 in Higham (2008).

O (2 \times n^{2})

of complex allocatable memory is required by the function.

If condition number and residual bound estimates are required, then nag_matop_complex_gen_matrix_cond_sqrt (f01kd) 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} 105 + 121 i & - 21 + 157 i & 42 + 18 i & - 4 - 2 i \\ 174 + 72 i & 28 + 236 i & 51 + 31 i & 16 - 6 i \\ 176 + 52 i & 37 + 177 i & 23 + 27 i & 25 + 13 i \\ - 9 + 125 i & - 111 + 67 i & - 8 + 30 i & 8 i \end{array}) .

Open in the MATLAB editor: f01fn_example

function f01fn_example


fprintf('f01fn example results\n\n');

% Principal square root of complex matrix A

a = [ 105 + 121i   -21 + 157i   42 + 18i   -4 -  2i;
      174 +  72i    28 + 236i   51 + 31i   16 -  6i;
      176 +  52i    37 + 177i   23 + 27i   25 + 13i;
       -9 + 125i  -111 +  67i   -8 + 30i         8i]; 

[as, ifail] = f01fn(a);

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

f01fn example results

Square root of A:
  10.0000 + 5.0000i   3.0000 + 6.0000i   2.0000 - 1.0000i  -1.0000 + 1.0000i
   7.0000 - 1.0000i   9.0000 +10.0000i   3.0000 + 0.0000i  -0.0000 - 1.0000i
   7.0000 - 4.0000i   5.0000 + 5.0000i   3.0000 + 3.0000i   4.0000 - 1.0000i
   2.0000 + 5.0000i  -2.0000 + 5.0000i  -1.0000 + 2.0000i   2.0000 - 0.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox