NAG FL Interface
f01kdf (complex_​gen_​matrix_​cond_​sqrt)

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1 Purpose

f01kdf computes an estimate of the relative condition number, κA1/2, and a bound on the relative residual, in the Frobenius norm, for the square root of a complex n×n matrix A. The principal square root, A1/2, of A is also returned.

2 Specification

Fortran Interface
Subroutine f01kdf ( n, a, lda, alpha, condsa, ifail)
Integer, Intent (In) :: n, lda
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (Out) :: alpha, condsa
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*)
C Header Interface
#include <nag.h>
void  f01kdf_ (const Integer *n, Complex a[], const Integer *lda, double *alpha, double *condsa, Integer *ifail)
The routine may be called by the names f01kdf or nagf_matop_complex_gen_matrix_cond_sqrt.

3 Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix square root, A1/2, of A is the unique square root with eigenvalues in the right half-plane.
The Fréchet derivative of a matrix function A1/2 in the direction of the matrix E is the linear function mapping E to L(A,E) such that
(A+E)1/2 - A1/2 - L(A,E) = o(A) .  
The absolute condition number is given by the norm of the Fréchet derivative which is defined by
L(A) := maxE0 L(A,E) E .  
The Fréchet derivative is linear in E and can, therefore, be written as
vec (L(A,E)) = K(A) vec(E) ,  
where the vec operator stacks the columns of a matrix into one vector, so that K(A) is n2×n2.
f01kdf uses Algorithm 3.20 from Higham (2008) to compute an estimate γ such that γ K(X) F . The quantity of γ provides a good approximation to L(A)F. The relative condition number, κA1/2, is then computed via
κA1/2 = L(A)F AF A1/2 F .  
κA1/2 is returned in the argument condsa.
A1/2 is computed using the algorithm described in Higham (1987). This is a 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 α is a measure of the stability of the relative residual (see Section 7). It is computed via
α= A 1/2 F 2 AF .  

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 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least n.
On entry: the n×n matrix A.
On exit: the n×n principal matrix square root A1/2. Alternatively, if ifail=1, contains an n×n non-principal square root of A.
3: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01kdf is called.
Constraint: ldan.
4: alpha Real (Kind=nag_wp) Output
On exit: an estimate of the stability of the relative residual for the computed principal (if ifail=0) or non-principal (if ifail=1) matrix square root, α.
5: condsa Real (Kind=nag_wp) Output
On exit: an estimate of the relative condition number, in the Frobenius norm, of the principal (if ifail=0) or non-principal (if ifail=1) matrix square root at A, κA1/2.
6: ifail Integer Input/Output
On entry: ifail must be set to 0, −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 −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 −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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
A has a negative or semisimple vanishing eigenvalue. A non-principal square root was returned.
A has a defective vanishing eigenvalue. The square root and condition number cannot be found in this case.
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.
An error occurred when computing the condition number. The matrix square root was still returned but you should use f01fnf to check if it is the principal matrix square root.
On entry, n=value.
Constraint: n0.
On entry, lda=value and n=value.
Constraint: ldan.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If the computed square root is X~, then the relative residual
A-X~2 F AF ,  
is bounded approximately by nαε, where ε is machine precision. The relative error in X~ is bounded approximately by nακA1/2ε.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01kdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kdf 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.

9 Further Comments

Approximately 3×n2 of complex allocatable memory is required by the routine.
The cost of computing the matrix square root is 85n3/3 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 f01fnf to obtain the matrix square root alone. Condition estimates for the square root of a real matrix can be obtained via f01jdf.

10 Example

This example estimates the matrix square root and condition number of the matrix
A = ( 29+35i 31+61i −38+49i −17-06i 52-59i 58-29i 97+39i −32+15i 20-31i 44-00i 37+19i −26+19i −70+72i −90+08i −87-43i 47-05i ) .  

10.1 Program Text

Program Text (f01kdfe.f90)

10.2 Program Data

Program Data (f01kdfe.d)

10.3 Program Results

Program Results (f01kdfe.r)