# NAG Library Routine Document

## 1Purpose

f01fnf computes the principal matrix square root, ${A}^{1/2}$, of a complex $n$ by $n$ matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f01fnf ( n, a, lda,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*)
#include nagmk26.h
 void f01fnf_ (const Integer *n, Complex a[], const Integer *lda, Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: 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 $A$.
3:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01fnf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
$A$ 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 routine was called incorrectly.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

f01fnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fnf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 routine.
If condition number and residual bound estimates are required, then f01kdf should be used. For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008).

## 10Example

This example finds the principal matrix square root of the matrix
 $A = 105+121i -21+157i 42+18i -4-02i 174+072i 28+236i 51+31i 16-06i 176+052i 37+177i 23+27i 25+13i -9+125i -111+067i -8+30i 08i .$

### 10.1Program Text

Program Text (f01fnfe.f90)

### 10.2Program Data

Program Data (f01fnfe.d)

### 10.3Program Results

Program Results (f01fnfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017