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 $n\times n$ matrix $A$. The principal square root, ${A}^{1/2}$, of $A$ is also returned.
The routine may be called by the names f01jdf or nagf_matop_real_gen_matrix_cond_sqrt.
3Description
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 half-plane.
The Fréchet derivative of a matrix function ${A}^{1/2}$ in the direction of the matrix $E$ is the linear function mapping $E$ to $L(A,E)$ such that
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
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 Science7782 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
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
2: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
${\mathbf{n}}$.
On entry: the $n\times n$ matrix $A$.
On exit: contains, if ${\mathbf{ifail}}={\mathbf{0}}$, the $n\times n$ principal matrix square root, ${A}^{1/2}$. Alternatively, if ${\mathbf{ifail}}={\mathbf{1}}$, contains an $n\times n$ 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 f01jdf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
4: $\mathbf{alpha}$ – 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 non-principal (if ${\mathbf{ifail}}={\mathbf{1}}$) matrix square root, $\alpha $.
5: $\mathbf{condsa}$ – 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 non-principal (if ${\mathbf{ifail}}={\mathbf{1}}$) matrix square root at $A$, ${\kappa}_{{A}^{1/2}}$.
6: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 $\mathrm{-1}$, 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 non-principal 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, non-principal 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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.
7Accuracy
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 $.
8Parallelism 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 implementation-specific information.
9Further Comments
Approximately $3\times {n}^{2}$ of real allocatable memory is required by the routine.
The cost of computing the matrix square root is $85{n}^{3}/3$ floating-point 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.
10Example
This example estimates the matrix square root and condition number of the matrix