NAG Library Routine Document
F01JDF
1 Purpose
F01JDF computes an estimate of the relative condition number ${\kappa}_{{A}^{1/2}}$ and a bound on the relative residual, in the Frobenius norm, for the square root of a real $n$ by $n$ matrix $A$. The principal square root, ${A}^{1/2}$, of $A$ is also returned.
2 Specification
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*), ALPHA, CONDSA 

3 Description
For a matrix with no eigenvalues on the closed negative real line, the principal matrix square root, ${A}^{1/2}$, of $A$ is the unique square root with eigenvalues in the right halfplane.
The Fréchet derivative of a matrix function
${A}^{1/2}$ in the direction of the matrix
$E$ is the linear function mapping
$E$ to
$L\left(A,E\right)$ such that
The absolute condition number is given by the norm of the Fréchet derivative which is defined by
The Fréchet derivative is linear in
$E$ and can therefore be written as
where the
$\mathrm{vec}$ operator stacks the columns of a matrix into one vector, so that
$K\left(A\right)$ is
${n}^{2}\times {n}^{2}$.
F01JDF uses Algorithm 3.20 from
Higham (2008) to compute an estimate
$\gamma $ such that
$\gamma \le {\Vert K\left(X\right)\Vert}_{F}$. The quantity of
$\gamma $ provides a good approximation to
${\Vert L\left(A\right)\Vert}_{F}$. The relative condition number,
${\kappa}_{{A}^{1/2}}$, is then computed via
${\kappa}_{{A}^{1/2}}$ is returned in the argument
CONDSA.
${A}^{1/2}$ 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.
The computed quantity
$\alpha $ is a measure of the stability of the relative residual (see
Section 7). It is computed via
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 $n$ 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: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01JDF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{ALPHA}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the stability of the relative residual for the computed principal (if ${\mathbf{IFAIL}}={\mathbf{0}}$) or nonprincipal (if ${\mathbf{IFAIL}}={\mathbf{1}}$) matrix square root, $\alpha $.
 5: $\mathrm{CONDSA}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the relative condition number, in the Frobenius norm, of the principal (if ${\mathbf{IFAIL}}={\mathbf{0}}$) or nonprincipal (if ${\mathbf{IFAIL}}={\mathbf{1}}$) matrix square root at $A$, ${\kappa}_{{A}^{1/2}}$.
 6: $\mathrm{IFAIL}$ – 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 a semisimple vanishing eigenvalue. A nonprincipal square root was returned.
 ${\mathbf{IFAIL}}=2$

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

$A$ has a negative real eigenvalue. The principal square root is not defined.
F01KDF can be used to return a complex, nonprincipal square root.
 ${\mathbf{IFAIL}}=4$

An error occurred when computing the matrix square root. Consequently,
ALPHA and
CONDSA could not be computed. It is likely that the routine was called incorrectly.
 ${\mathbf{IFAIL}}=5$

An error occurred when computing the condition number. The matrix square root was still returned but you should use
F01ENF to check if it is the principal matrix square root.
 ${\mathbf{IFAIL}}=1$

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
If the computed square root is
$\stackrel{~}{X}$, then the relative residual
is bounded approximately by
$n\alpha \epsilon $, where
$\epsilon $ is
machine precision. The relative error in
$\stackrel{~}{X}$ is bounded approximately by
$n\alpha {\kappa}_{{A}^{1/2}}\epsilon $.
8 Parallelism and Performance
F01JDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01JDF 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.
Approximately $3\times {n}^{2}$ of real allocatable memory is required by the routine.
The cost of computing the matrix square root is $85{n}^{3}/3$ floatingpoint operations. The cost of computing the condition number depends on how fast the algorithm converges. It typically takes over twice as long as computing the matrix square root.
If condition estimates are not required then it is more efficient to use
F01ENF to obtain the matrix square root alone. Condition estimates for the square root of a complex matrix can be obtained via
F01KDF.
10 Example
This example estimates the matrix square root and condition number of the matrix
10.1 Program Text
Program Text (f01jdfe.f90)
10.2 Program Data
Program Data (f01jdfe.d)
10.3 Program Results
Program Results (f01jdfe.r)