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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_complex_gen_matrix_sqrt (f01fn)

## Purpose

nag_matop_complex_gen_matrix_sqrt (f01fn) computes the principal matrix square root, ${A}^{1/2}$, of a complex $n$ by $n$ matrix $A$.

## Syntax

[a, ifail] = f01fn(a, 'n', n)
[a, ifail] = nag_matop_complex_gen_matrix_sqrt(a, 'n', n)

## Description

A square root of a matrix $A$ is a solution $X$ to the equation ${X}^{2}=A$. 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least ${\mathbf{n}}$.
The second dimension of the array a must be at least ${\mathbf{n}}$.
The $n$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be ${\mathbf{n}}$.
The second dimension of the array a will be ${\mathbf{n}}$.
Contains, if ${\mathbf{ifail}}={\mathbf{0}}$, the $n$ by $n$ principal matrix square root, ${A}^{1/2}$. Alternatively, if ${\mathbf{ifail}}={\mathbf{1}}$, contains an $n$ by $n$ non-principal square root of $A$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
$A$ has a negative or semisimple vanishing eigenvalue. A non-principal square root is returned.
${\mathbf{ifail}}=2$
$A$ has a defective vanishing eigenvalue. The square root cannot be found in this case.
${\mathbf{ifail}}=3$
An internal error occurred. It is likely that the function was called incorrectly.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
Constraint: $\mathit{lda}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The computed square root $\stackrel{^}{X}$ satisfies ${\stackrel{^}{X}}^{2}=A+\Delta A$, where ${‖\Delta A‖}_{F}\approx O\left(\epsilon \right){n}^{3}{‖\stackrel{^}{X}‖}_{F}^{2}$, where $\epsilon$ is machine precision.

The cost of the algorithm is $85{n}^{3}/3$ complex floating-point operations; see Algorithm 6.3 in Higham (2008). $O\left(2×{n}^{2}\right)$ 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
 $A = 105+121i -21+157i 42+18i -4-02i 174+072i 28+236i 51+31i 16-06i 176+052i 37+177i 23+27i 25+13i -9+125i -111+067i -8+30i 08i .$
```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

```