NAG Library Function Document
nag_matop_complex_tri_matrix_sqrt (f01fpc) computes the principal matrix square root, , of a complex upper triangular by matrix .
||nag_matop_complex_tri_matrix_sqrt (Integer n,
A square root of a matrix is a solution to the equation . 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 , is the unique square root whose eigenvalues lie in the open right half-plane.
nag_matop_complex_tri_matrix_sqrt (f01fpc) computes , where is an upper triangular matrix. is also upper triangular.
The algorithm used by nag_matop_complex_tri_matrix_sqrt (f01fpc) is described in Björck and Hammarling (1983)
. In addition a blocking scheme described in Deadman et al. (2013)
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
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the by upper triangular matrix .
: contains, if
principal matrix square root,
. Alternatively, if NE_EIGENVALUES
, contains an
non-principal square root of
pda – IntegerInput
: the stride separating matrix row elements in the array a
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
has negative or semisimple, vanishing eigenvalues. The principal square root is not defined in this case; a non-principal square root is returned.
On entry, .
On entry, and .
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
has a defective vanishing eigenvalue. The square root cannot be found in this case.
The computed square root satisfies , where , where is machine precision. The order of the change in is to be interpreted elementwise.
8 Parallelism and Performance
nag_matop_complex_tri_matrix_sqrt (f01fpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_tri_matrix_sqrt (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 Users' Note
for your implementation for any additional implementation-specific information.
The cost of the algorithm is
complex floating-point operations; see Algorithm 6.3 in Higham (2008)
of complex allocatable memory is required by the function.
is a full matrix, then nag_matop_complex_gen_matrix_sqrt (f01fnc)
should be used to compute the principal square root.
If condition number and residual bound estimates are required, then nag_matop_complex_gen_matrix_cond_sqrt (f01kdc)
should be used. For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008)
This example finds the principal matrix square root of the matrix
10.1 Program Text
Program Text (f01fpce.c)
10.2 Program Data
Program Data (f01fpce.d)
10.3 Program Results
Program Results (f01fpce.r)