NAG Library Routine Document
f01enf (real_gen_matrix_sqrt)
1
Purpose
f01enf computes the principal matrix square root, ${A}^{1/2}$, of a real $n$ by $n$ matrix $A$.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n, lda  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (Inout)  ::  a(lda,*) 

C Header Interface
#include nagmk26.h
void 
f01enf_ (const Integer *n, double a[], const Integer *lda, Integer *ifail) 

3
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 halfplane.
${A}^{1/2}$ 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.
4
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
5
Arguments
 1: $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 2: $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (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 $n$ nonprincipal 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
f01enf 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).
6
Error 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 semisimple vanishing eigenvalue. A nonprincipal square root is returned.
 ${\mathbf{ifail}}=2$

$A$ has a defective vanishing eigenvalue. The square root cannot be found in this case.
 ${\mathbf{ifail}}=3$

$A$ has a negative real eigenvalue. The principal square root is not defined.
f01fnf can be used to return a complex, nonprincipal square root.
 ${\mathbf{ifail}}=4$

An internal error occurred. It is likely that the routine was called incorrectly.
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
7
Accuracy
The computed square root $\hat{X}$ satisfies ${\hat{X}}^{2}=A+\Delta A$, where ${\Vert \Delta A\Vert}_{F}\approx O\left(\epsilon \right){n}^{3}{\Vert \hat{X}\Vert}_{F}^{2}$, where $\epsilon $ is machine precision.
For further discussion of the condition of the matrix square root see Section 6.1 of
Higham (2008).
8
Parallelism and Performance
f01enf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01enf 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 implementationspecific information.
The cost of the algorithm is
$85{n}^{3}/3$ floatingpoint operations; see Algorithm 6.7 of
Higham (2008).
$O\left(2\times {n}^{2}\right)$ of real allocatable memory is required by the routine.
If condition number and residual bound estimates are required, then
f01jdf should be used.
10
Example
This example finds the principal matrix square root of the matrix
10.1
Program Text
Program Text (f01enfe.f90)
10.2
Program Data
Program Data (f01enfe.d)
10.3
Program Results
Program Results (f01enfe.r)