NAG CL Interfacef01fpc (complex_​tri_​matrix_​sqrt)

1Purpose

f01fpc computes the principal matrix square root, ${A}^{1/2}$, of a complex upper triangular $n$ by $n$ matrix $A$.

2Specification

 #include
 void f01fpc (Integer n, Complex a[], Integer pda, NagError *fail)
The function may be called by the names: f01fpc or nag_matop_complex_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.
f01fpc computes ${A}^{1/2}$, where $A$ is an upper triangular matrix. ${A}^{1/2}$ is also upper triangular.
The algorithm used by f01fpc is described in Björck and Hammarling (1983). In addition a blocking scheme described in Deadman et al. (2013) is used.

4References

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

5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ upper triangular matrix $A$.
On exit: contains, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the $n$ by $n$ principal matrix square root, ${A}^{1/2}$. Alternatively, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES, contains an $n$ by $n$ non-principal square root of $A$.
3: $\mathbf{pda}$Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENVALUES
$A$ has negative or semisimple, vanishing eigenvalues. The principal square root is not defined in this case; a non-principal square root is returned.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
$A$ has a defective vanishing eigenvalue. The square root cannot be found in this case.

7Accuracy

The computed square root $\stackrel{^}{X}$ satisfies ${\stackrel{^}{X}}^{2}=A+\Delta A$, where $\left|\Delta A\right|\approx O\left(\epsilon \right)n{\left|\stackrel{^}{X}\right|}^{2}$, where $\epsilon$ is machine precision. The order of the change in $A$ is to be interpreted elementwise.

8Parallelism and Performance

f01fpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fpc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The cost of the algorithm is ${n}^{3}/3$ complex floating-point operations; see Algorithm 6.3 in Higham (2008). $O\left(2×{n}^{2}\right)$ of complex allocatable memory is required by the function.
If $A$ is a full matrix, then f01fnc should be used to compute the principal square root.
If condition number and residual bound estimates are required, then f01kdc 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
 $A = 2i 14+02i 12+03i 6+04i 0i+0 -5+12i 6+18i 9+16i 0i+0 0i+00 3-04i 16-04i 0i+0 0i+00 0i+00 4i+00 .$

10.1Program Text

Program Text (f01fpce.c)

10.2Program Data

Program Data (f01fpce.d)

10.3Program Results

Program Results (f01fpce.r)