generateTestBandComplex¶

Generates test matrices of class \(E_c(n,c)\). Returns in band or band symmetric format.

Synopsis¶

generateTestBandComplex (n, c)

Required Arguments¶

int n (Input): Number of rows in the matrix.
int c (Input): Parameter used to alter structure, also the number of upper/lower codiagonals

Return Value¶

A vector of type f_complex. If no test was generated, then None is returned.

Optional Arguments¶

symmetricStorage,: Return matrix stored in band symmetric format.

Description¶

We use the same nomenclature as Østerby and Zlatev (1982). Test matrices of class \(E(n,c)\), to which we will generally refer to as E-matrices, are symmetric, positive definite matrices of order n with (6.0, 0.0) in the diagonal, (-1.0, 1.0) in the superdiagonal and (-1.0, -1.0) subdiagonal. In addition there are two bands at a distance c from the diagonal with (-1.0, 1.0) in the upper codiagonal and (-1.0, − 1.0) in the lower codiagonal. More precisely:

\(a_{i,i}=6\)	0 ≤ i < `n`
\(a_{i,i+1}=-1-i\)	0 ≤ i < `n` −1
\(a_{i+1,}i=-1-i\)	0 ≤ i < `n` − 1
\(a_{i,i+c}=-1+i\)	0 ≤ i < `n` − c
\(a_{i+c,i}=-1+i\)	0 ≤ i < `n` − c

for any \(n\geq 3\) and \(2\leq c\leq n-1\).

E-matrices are similar to those obtained from the five-point formula in the discretization of elliptic partial differential equations.

By default, generateTestBandComplex returns an E-matrix in band storage mode. Option symmetricStorage returns a matrix in band symmetric storage mode.

Example¶

This example generates the following matrix and prints the result:

\[\begin{split}E_c(5,3) = \begin{bmatrix} 6 & -1-i & 0 & -1+i & 0 \\ -1-i & 6 & -1+i & 0 & -1+i \\ 0 & -1-i & 6 & -1+i & 0 \\ -1-i & 0 & -1-i & 6 & -1+i \\ 0 & -1-i & 0 & -1-i & 6 \\ \end{bmatrix}\end{split}\]

from pyimsl.math.generateTestBandComplex import generateTestBandComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

a = generateTestBandComplex(5, 3)

writeMatrixComplex("E(5, 3) in band storage: ", a)

Output¶

 
               E(5, 3) in band storage: 
                           1                          2
(          0,          0)  (          0,          0)
(          0,          0)  (          0,          0)
(          0,          0)  (         -1,          1)
(          6,          0)  (          6,          0)
(         -1,         -1)  (         -1,         -1)
(          0,          0)  (          0,          0)
(         -1,         -1)  (         -1,         -1)
 
                           3                          4
(          0,          0)  (         -1,          1)
(          0,          0)  (          0,          0)
(         -1,          1)  (         -1,          1)
(          6,          0)  (          6,          0)
(         -1,         -1)  (         -1,         -1)
(          0,          0)  (          0,          0)
(          0,          0)  (          0,          0)
 
                           5
(         -1,          1)
(          0,          0)
(         -1,          1)
(          6,          0)
(          0,          0)
(          0,          0)
(          0,          0)