NAG Library Routine Document

f01kbf (complex_gen_matrix_cond_num)


    1  Purpose
    7  Accuracy


f01kbf computes an estimate of the absolute condition number of a matrix function f of a complex n by n matrix A in the 1-norm. Numerical differentiation is used to evaluate the derivatives of f when they are required.


Fortran Interface
Subroutine f01kbf ( n, a, lda, f, iuser, ruser, iflag, conda, norma, normfa, ifail)
Integer, Intent (In):: n, lda
Integer, Intent (Inout):: iuser(*), ifail
Integer, Intent (Out):: iflag
Real (Kind=nag_wp), Intent (Inout):: ruser(*)
Real (Kind=nag_wp), Intent (Out):: conda, norma, normfa
Complex (Kind=nag_wp), Intent (Inout):: a(lda,*)
External:: f
C Header Interface
#include nagmk26.h
void  f01kbf_ (const Integer *n, Complex a[], const Integer *lda,
void (NAG_CALL *f)(Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[]),
Integer iuser[], double ruser[], Integer *iflag, double *conda, double *norma, double *normfa, Integer *ifail)


The absolute condition number of f at A, condabsf,A is given by the norm of the Fréchet derivative of f, LA, which is defined by
LX := maxE0 LX,E E ,  
where LX,E is the Fréchet derivative in the direction E. LX,E is linear in E and can therefore be written as
vec LX,E = KX vecE ,  
where the vec operator stacks the columns of a matrix into one vector, so that KX is n2×n2. f01kbf computes an estimate γ such that γ KX 1 , where KX 1 n-1 LX 1 , n LX 1 . The relative condition number can then be computed via
cond rel f,A = cond abs f,A A1 fA 1 .  
The algorithm used to find γ is detailed in Section 3.4 of Higham (2008).


Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA


1:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     alda* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least n.
On entry: the n by n matrix A.
On exit: the n by n matrix, fA.
3:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01kbf is called.
Constraint: ldan.
4:     f – Subroutine, supplied by the user.External Procedure
The subroutine f evaluates fzi at a number of points zi.
The specification of f is:
Fortran Interface
Subroutine f ( iflag, nz, z, fz, iuser, ruser)
Integer, Intent (In):: nz
Integer, Intent (Inout):: iflag, iuser(*)
Real (Kind=nag_wp), Intent (Inout):: ruser(*)
Complex (Kind=nag_wp), Intent (In):: z(nz)
Complex (Kind=nag_wp), Intent (Out):: fz(nz)
C Header Interface
#include nagmk26.h
void  f (Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[])
1:     iflag – IntegerInput/Output
On entry: iflag will be zero.
On exit: iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function fz; for instance fz may not be defined. If iflag is returned as nonzero then f01kbf will terminate the computation, with ifail=3.
2:     nz – IntegerInput
On entry: nz, the number of function values required.
3:     znz – Complex (Kind=nag_wp) arrayInput
On entry: the nz points z1,z2,,znz at which the function f is to be evaluated.
4:     fznz – Complex (Kind=nag_wp) arrayOutput
On exit: the nz function values. fzi should return the value fzi, for i=1,2,,nz.
5:     iuser* – Integer arrayUser Workspace
6:     ruser* – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to f01kbf. You should use the arrays iuser and ruser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f01kbf is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01kbf. If your code inadvertently does return any NaNs or infinities, f01kbf is likely to produce unexpected results.
5:     iuser* – Integer arrayUser Workspace
6:     ruser* – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by f01kbf, but are passed directly to f and may be used to pass information to this routine.
7:     iflag – IntegerOutput
On exit: iflag=0, unless iflag has been set nonzero inside f, in which case iflag will be the value set and ifail will be set to ifail=3.
8:     conda – Real (Kind=nag_wp)Output
On exit: an estimate of the absolute condition number of f at A.
9:     norma – Real (Kind=nag_wp)Output
On exit: the 1-norm of A.
10:   normfa – Real (Kind=nag_wp)Output
On exit: the 1-norm of fA.
11:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ 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 argument, the recommended value is 0. 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).

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:
An internal error occurred when estimating the norm of the Fréchet derivative of f at A. Please contact NAG.
An internal error occurred while evaluating the matrix function fA. You can investigate further by calling f01flf with the matrix A and the function f.
iflag has been set nonzero by the user-supplied subroutine.
On entry, n<0.
On entry, argument lda is invalid.
Constraint: ldan.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


f01kbf uses the norm estimation routine f04zdf to estimate a quantity γ, where γ KX 1  and KX 1 n-1 LX 1 , n LX 1 . For further details on the accuracy of norm estimation, see the documentation for f04zdf.

Parallelism and Performance

f01kbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this routine may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation. The user workspace arrays iuser and ruser are classified as OpenMP shared memory and use of iuser and ruser has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays. If at all possible, it is recommended that these arrays are only used to supply read-only data to the user functions when a multithreaded implementation is being used.
f01kbf 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.

Further Comments

Approximately 6n2 of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine f01flf.
f01kbf returns the matrix function fA. This is computed using f01flf. If only fA is required, without an estimate of the condition number, then it is far more efficient to use f01flf directly.
The real analogue of this routine is f01jbf.


This example estimates the absolute and relative condition numbers of the matrix function sin2A where
A = 2.0+0.0i 0.0+1.0i 1.0+1.0i 0.0+3.0i 1.0+1.0i 0.0+2.0i 2.0+2.0i 0.0+0.0i 0.0+0.0i 2.0+0.0i 1.0+2.0i 1.0+0.0i 1.0+1.0i 3.0+0.0i 0.0+0.0i 1.0+2.0i .  

Program Text

Program Text (f01kbfe.f90)

Program Data

Program Data (f01kbfe.d)

Program Results

Program Results (f01kbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017