matMulRectBandComplex¶

Computes the transpose of a matrix, a matrix-vector product, or a matrix-matrix product for all matrices of complex type and stored in band form.

Synopsis¶

matMulRectBandComplex (string)

Required Arguments¶

char string (Input): String indicating matrix multiplication to be performed.

Return Value¶

The result of the multiplication is returned.

Optional Arguments¶

aMatrix, int nrowa, int ncola, int nlca, int nuca, complex a (Input): The sparse matrix

\[A \in \mathbb{C}^{\mathrm{nrowa} \times \mathrm{ncola}}\]

bMatrix, int nrowb, int ncolb, int nlcb, int nucb, complex b (Input): The sparse matrix

\[B \in \mathbb{C}^{\mathrm{nrowb} \times \mathrm{ncolb}}\]

xVector, int nx, complex x, (Input): The vector x of length nx.
returnMatrixCodiagonals, int nlc_result, int nuc_result, (Output): If the function matMulRectBandComplex returns data for a band matrix, use this option to retrieve the number of lower and upper codiagonals of the return matrix.

Description¶

The function matMulRectBandComplex computes a matrix-matrix product or a matrix‑vector product, where the matrices are specified in band format. The operation performed is specified by string. For example, if “A*x” is given, Ax is computed. In string, the matrices A and B and the vector x can be used. Any of these names can be used with trans, indicating transpose. The vector x is treated as a dense n × 1 matrix. If string contains only one item, such as “x” or “trans(A)”, then a copy of the array, or its transpose is returned.

The matrices and/or vector referred to in string must be given as optional arguments. Therefore, if string is “A*x”, then aMatrix and xVector must be given.

Examples¶

Example 1¶

Let

\[\begin{split}A = \begin{bmatrix} -2 & 4 & 0 & 0 \\ 6+i & -0.5+3i & -2+2i & 0 \\ 0 & 1+i & 3-3i & -4-i \\ 0 & 0 & 2i & 1-i \\ \end{bmatrix}\end{split}\]

and

\[\begin{split}x = \begin{bmatrix} 3 \\ -1+i \\ 3 \\ -1+i \\ \end{bmatrix}\end{split}\]

This example computes the product Ax.

from pyimsl.math.matMulRectBandComplex import matMulRectBandComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

n = 4
nlca = 1
nuca = 1

# Note that a is in band storage mode
a = [[0 + 0j, 4 + 0j, -2 + 2j, -4 - 1j],
     [-2 - 3j, -.5 + 3j, 3 - 3j, 1 - 1j],
     [6 + 1j, 1 + 1j, 0 + 2j, 0 + 0j]]
x = [3 + 0j, -1 + 1j, 3 + 0j, -1 + 1j]

# Set b = A*x
b = matMulRectBandComplex(
    "A*x", aMatrix={'nrowa': n, 'ncola': n, 'nlca': nlca, 'nuca': nuca, 'a': a}, xVector=x)

writeMatrixComplex("Product, Ax", b)

Output¶

 
                     Product, Ax
                        1                          2
(      -10.0,       -5.0)  (        9.5,        5.5)
 
                        3                          4
(       12.0,      -12.0)  (        0.0,        8.0)

Example 2¶

Using the same matrix A and vector x given in the last example, the products Ax, \(A^Tx\), \(A^Hx\) and \(AA^H\) are computed.

from pyimsl.math.matMulRectBandComplex import matMulRectBandComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

n = 4
nlca = 1
nuca = 1

# Note that a is in band storage mode
a = [[0 + 0j, 4 + 0j, -2 + 2j, -4 - 1j],
     [-2 - 3j, -.5 + 3j, 3 - 3j, 1 - 1j],
     [6 + 1j, 1 + 1j, 0 + 2j, 0 + 0j]]
x = [3 + 0j, -1 + 1j, 3 + 0j, -1 + 1j]

# Set b = A*x
b = matMulRectBandComplex(
    "A*x", aMatrix={'nrowa': n, 'ncola': n, 'nlca': nlca, 'nuca': nuca, 'a': a}, xVector=x)
writeMatrixComplex("Product, Ax", b)

# Set b = trans(A)*x
b = matMulRectBandComplex(
    "trans(A)*x", aMatrix={'nrowa': n, 'ncola': n, 'nlca': nlca, 'nuca': nuca, 'a': a}, xVector=x)
writeMatrixComplex("trans(A)*x", b)

# Set b = ctrans(A)*x
b = matMulRectBandComplex(
    "ctrans(A)*x", aMatrix={'nrowa': n, 'ncola': n, 'nlca': nlca, 'nuca': nuca, 'a': a}, xVector=x)
writeMatrixComplex("ctrans(A)*x", b)

# Set z = A*ctrans(A
codiags = []
b = matMulRectBandComplex("A*ctrans(A)", aMatrix={'nrowa': n, 'ncola': n,
                                                  'nlca': nlca, 'nuca': nuca, 'a': a}, returnMatrixCodiagonals=codiags)
writeMatrixComplex("A*ctrans(A)", b)

Output¶

 
                     Product, Ax
                        1                          2
(      -10.0,       -5.0)  (        9.5,        5.5)
 
                        3                          4
(       12.0,      -12.0)  (        0.0,        8.0)
 
                     trans(A)*x
                        1                          2
(      -13.0,       -4.0)  (       12.5,       -0.5)
 
                        3                          4
(        7.0,      -15.0)  (      -12.0,       -1.0)
 
                     ctrans(A)*x
                        1                          2
(      -11.0,       16.0)  (       18.5,       -0.5)
 
                        3                          4
(       15.0,       11.0)  (      -14.0,        3.0)
 
                      A*ctrans(A)
                           1                          2
1  (       0.00,       0.00)  (       0.00,       0.00)
2  (       0.00,       0.00)  (     -17.00,     -28.00)
3  (      29.00,       0.00)  (      54.25,       0.00)
4  (     -17.00,      28.00)  (      -9.50,      -3.50)
5  (       4.00,       4.00)  (       4.00,      -4.00)
 
                           3                          4
1  (       4.00,      -4.00)  (       4.00,       4.00)
2  (      -9.50,       3.50)  (      -9.00,     -11.00)
3  (      37.00,       0.00)  (       6.00,       0.00)
4  (      -9.00,      11.00)  (       0.00,       0.00)
5  (       0.00,       0.00)  (       0.00,       0.00)