NAG Library Routine Document
F01EPF
1 Purpose
F01EPF computes the principal matrix square root, ${A}^{1/2}$, of a real upper quasitriangular $n$ by $n$ matrix $A$.
2 Specification
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*) 

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.
F01EPF computes
${A}^{1/2}$, where
$A$ is an upper quasitriangular matrix, with
$1\times 1$ and
$2\times 2$ blocks on the diagonal. Such matrices arise from the Schur factorization of a real general matrix, as computed by
F08PEF (DHSEQR), for example. F01EPF does not require
$A$ to be in the canonical Schur form described in
F08PEF (DHSEQR), it merely requires
$A$ to be upper quasitriangular.
${A}^{1/2}$ then has the same block triangular structure as
$A$.
The algorithm used by F01EPF is described in
Higham (1987). 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 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – 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 $n$ upper quasitriangular matrix $A$.
On exit: the $n$ by $n$ principal matrix square root ${A}^{1/2}$.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01EPF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction 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 parameter, 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
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, 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 vanishing eigenvalues. The principal square root is not defined in this case.
F01ENF or
F01FNF may be able to provide further information.
 ${\mathbf{IFAIL}}=2$

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 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
The computed square root $\hat{X}$ satisfies ${\hat{X}}^{2}=A+\Delta A$, where ${\Vert \Delta A\Vert}_{F}\approx O\left(\epsilon \right)n{\Vert \hat{X}\Vert}_{F}^{2}$, where $\epsilon $ is machine precision.
8 Parallelism and Performance
F01EPF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01EPF 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
${n}^{3}/3$ floatingpoint operations; see Algorithm 6.7 of
Higham (2008).
$O\left(n\right)$ of integer allocatable memory is required by the routine.
If
$A$ is a full matrix, then
F01ENF should be used to compute the square root. If
$A$ has negative real eigenvalues then
F01FNF can be used to return a complex, nonprincipal square root.
If condition number and residual bound estimates are required, then
F01JDF 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
Program Text (f01epfe.f90)
10.2 Program Data
Program Data (f01epfe.d)
10.3 Program Results
Program Results (f01epfe.r)