NAG FL Interface
f01fpf (complex_tri_matrix_sqrt)
1
Purpose
f01fpf computes the principal matrix square root, ${A}^{1/2}$, of a complex upper triangular $n$ by $n$ matrix $A$.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, lda 
Integer, Intent (Inout) 
:: 
ifail 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*) 

C Header Interface
#include <nag.h>
void 
f01fpf_ (const Integer *n, Complex a[], const Integer *lda, Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f01fpf_ (const Integer &n, Complex a[], const Integer &lda, Integer &ifail) 
}

The routine may be called by the names f01fpf or nagf_matop_complex_tri_matrix_sqrt.
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.
f01fpf computes ${A}^{1/2}$, where $A$ is an upper triangular matrix. ${A}^{1/2}$ is also upper triangular.
The algorithm used by
f01fpf is described in
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 (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

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) array
Input/Output

Note: the second dimension of the array
a
must be at least
${\mathbf{n}}$.
On entry: the $n$ by $n$ upper triangular 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:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f01fpf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.

4:
$\mathbf{ifail}$ – Integer
Input/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 4 in the Introduction to the NAG Library FL Interface 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 negative or semisimple, vanishing eigenvalues. The principal square root is not defined in this case; a nonprincipal square root is returned.
 ${\mathbf{ifail}}=2$

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

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 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
The computed square root $\hat{X}$ satisfies ${\hat{X}}^{2}=A+\Delta A$, where $\left\Delta A\right\approx O\left(\epsilon \right)n{\left\hat{X}\right}^{2}$, where $\epsilon $ is machine precision. The order of the change in $A$ is to be interpreted elementwise.
8
Parallelism and Performance
f01fpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fpf 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
${n}^{3}/3$ complex floatingpoint operations; see Algorithm 6.3 in
Higham (2008).
$O\left(2\times {n}^{2}\right)$ of complex allocatable memory is required by the routine.
If
$A$ is a full matrix, then
f01fnf should be used to compute the principal square root.
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).
10
Example
This example finds the principal matrix square root of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results