NAG Library Function Document
nag_matop_real_tri_matrix_sqrt (f01epc) computes the principal matrix square root, , of a real upper quasi-triangular by matrix .
||nag_matop_real_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_real_tri_matrix_sqrt (f01epc) computes
is an upper quasi-triangular matrix, with
blocks on the diagonal. Such matrices arise from the Schur factorization of a real general matrix, as computed by nag_dhseqr (f08pec)
, for example. nag_matop_real_tri_matrix_sqrt (f01epc) does not require
to be in the canonical Schur form described in nag_dhseqr (f08pec)
, it merely requires
to be upper quasi-triangular.
then has the same block triangular structure as
The algorithm used by nag_matop_real_tri_matrix_sqrt (f01epc) is described in Higham (1987)
. 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 (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
n – IntegerInput
On entry: , the order of the matrix .
a – doubleInput/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 quasi-triangular matrix .
On exit: the by principal matrix square root .
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 vanishing eigenvalues. The principal square root is not defined in this case. nag_matop_real_gen_matrix_sqrt (f01enc)
or nag_matop_complex_gen_matrix_sqrt (f01fnc)
may be able to provide further information.
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
The computed square root satisfies , where , where is machine precision.
8 Parallelism and Performance
nag_matop_real_tri_matrix_sqrt (f01epc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_tri_matrix_sqrt (f01epc) 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
floating-point operations; see Algorithm 6.7 of Higham (2008)
of integer allocatable memory is required by the function.
is a full matrix, then nag_matop_real_gen_matrix_sqrt (f01enc)
should be used to compute the square root. If
has negative real eigenvalues then nag_matop_complex_gen_matrix_sqrt (f01fnc)
can be used to return a complex, non-principal square root.
If condition number and residual bound estimates are required, then nag_matop_real_gen_matrix_cond_sqrt (f01jdc)
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 (f01epce.c)
10.2 Program Data
Program Data (f01epce.d)
10.3 Program Results
Program Results (f01epce.r)