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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_real_tri_matrix_sqrt (f01ep)

## Purpose

nag_matop_real_tri_matrix_sqrt (f01ep) computes the principal matrix square root, ${A}^{1/2}$, of a real upper quasi-triangular $n$ by $n$ matrix $A$.

## Syntax

[a, ifail] = f01ep(a, 'n', n)
[a, ifail] = nag_matop_real_tri_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.
nag_matop_real_tri_matrix_sqrt (f01ep) computes ${A}^{1/2}$, where $A$ is an upper quasi-triangular matrix, with $1×1$ and $2×2$ 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $n$ upper quasi-triangular 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be ${\mathbf{n}}$.
The second dimension of the array a will be ${\mathbf{n}}$.
The $n$ by $n$ principal matrix square root ${A}^{1/2}$.
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 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.
${\mathbf{ifail}}=2$
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 ${‖\Delta A‖}_{F}\approx O\left(\epsilon \right)n{‖\stackrel{^}{X}‖}_{F}^{2}$, where $\epsilon$ is machine precision.

The cost of the algorithm is ${n}^{3}/3$ floating-point operations; see Algorithm 6.7 of Higham (2008). $O\left(n\right)$ of integer allocatable memory is required by the function.
If $A$ 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
 $A = 6 4 -5 15 8 6 -3 10 0 0 3 -4 0 0 4 3 .$
```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

```