The routine may be called by the names f01epf or nagf_matop_real_tri_matrix_sqrt.
3Description
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 half-plane.
f01epf computes ${A}^{1/2}$, where $A$ is an upper quasi-triangular 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, for example. f01epf does not require $A$ to be in the canonical Schur form described in f08pef, it merely requires $A$ to be upper quasi-triangular. ${A}^{1/2}$ then has the same block triangular structure as $A$.
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 $n\times n$ upper quasi-triangular matrix $A$.
On exit: the $n\times n$ principal matrix square root ${A}^{1/2}$.
3: $\mathbf{lda}$ – 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: $\mathbf{ifail}$ – 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 negative or vanishing eigenvalues. The principal square root is not defined in this case. f01enforf01fnf 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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 implementation-specific information.
9Further Comments
The cost of the algorithm is ${n}^{3}/3$ floating-point 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, non-principal 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).
10Example
This example finds the principal matrix square root of the matrix