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3.3 Level 2 BLAS

3.3 Level 2 BLAS The level 2 functions implement matrix-vector products and rank-1 and rank-2 matrix updates. Different types of matrix structure can be exploited using the conventions of section 3.1.

gemv(A, x, y[, trans='N'[, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a general matrix:

$\begin{displaymath} y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'... ...\alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}). \end{displaymath}$

The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

symv(A, x, y[, uplo='L'[, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a real symmetric matrix:

$\begin{displaymath} y := \alpha A x + \beta y, \end{displaymath}$

where A is a real symmetric matrix. The arguments A, x and y must have type 'd' and alpha and beta must be real.

hemv(A, x, y[, uplo='L'[, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a real symmetric or complex Hermitian matrix:

$\begin{displaymath} y := \alpha A x + \beta y, \end{displaymath}$

where A is real symmetric or complex Hermitian. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

trmv(A, x[, uplo='L'[, trans='N'[, diag='N']]])

Matrix-vector product with a triangular matrix:

$\begin{displaymath} x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad x := A... ...'}), \qquad x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}), \end{displaymath}$

where A is square and triangular. The arguments A and x must have the same type ('d' or 'z').

trsv(A, x[, uplo='L'[, trans='N'[, diag='N']]])

Solution of a nonsingular triangular set of linear equations:

$\begin{displaymath} x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad x... ..., \qquad x := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}), \end{displaymath}$

where A is square and triangular with nonzero diagonal elements. The arguments A and x must have the same type ('d' or 'z').

gbmv(A, m, kl, x, y[, trans='N' [, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a general band matrix:

$\begin{displaymath} y := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'... ... \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}), \end{displaymath}$

where A is a rectangular band matrix with m rows and k_l subdiagonals. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

sbmv(A, x, y[, uplo='L'[, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a real symmetric band matrix:

$\begin{displaymath} y := \alpha A x + \beta y, \end{displaymath}$

where A is a real symmetric band matrix. The arguments A, x and y must have type 'd' and alpha and beta must be real.

hbmv(A, x, y[, uplo='L'[, alpha=1.0[, beta=0.0]]])

Matrix-vector product with a real symmetric or complex Hermitian band matrix:

$\begin{displaymath} y := \alpha A x + \beta y, \end{displaymath}$

where A is a real symmetric or complex Hermitian band matrix. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

tbmv(A, x[, uplo='L'[, trans[, diag]]])

Matrix-vector product with a triangular band matrix:

$\begin{displaymath} x := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \qquad x := A... ...}), \qquad x := A^H x \quad (\mathrm{trans} = \mathrm{'C'}). \end{displaymath}$

The arguments A and x must have the same type ('d' or 'z').

tbsv(A, x[, uplo='L'[, trans[, diag]]])

Solution of a triangular banded set of linear equations:

$\begin{displaymath} x := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \qquad x... ..., \qquad x := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}), \end{displaymath}$

where A is a triangular band matrix of with nonzero diagonal elements. The arguments A and x must have the same type ('d' or 'z').

ger(x, y, A[, alpha=1.0])

General rank-1 update:

$\begin{displaymath} A := A + \alpha x y^H, \end{displaymath}$

where A is a general matrix. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

geru(x, y, A[, alpha=1.0])

General rank-1 update:

$\begin{displaymath} A := A + \alpha x y^T, \end{displaymath}$

where A is a general matrix. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

syr(x, A[, uplo='L'[, alpha=1.0]])

Symmetric rank-1 update:

$\begin{displaymath} A := A + \alpha xx^T, \end{displaymath}$

where A is a real symmetric matrix. The arguments A and x must have type 'd'. alpha must be a real number.

her(x, A[, uplo='L'[, alpha=1.0]])

Hermitian rank-1 update:

$\begin{displaymath} A := A + \alpha xx^H, \end{displaymath}$

where A is a real symmetric or complex Hermitian matrix. The arguments A and x must have the same type ('d' or 'z'). alpha must be a real number.

syr2(x, y, A[, uplo='L'[, alpha=1.0]])

Symmetric rank-2 update:

$\begin{displaymath} A := A + \alpha (xy^T + yx^T), \end{displaymath}$

where A is a real symmetric matrix. The arguments A, x and y must have type 'd'. alpha must be real.

her2(x, y, A[, uplo='L'[, alpha=1.0]])

Symmetric rank-2 update:

$\begin{displaymath} A := A + \alpha xy^H + \bar \alpha yx^H, \end{displaymath}$

where A is a a real symmetric or complex Hermitian matrix. The arguments A, x and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

As an example, the following code multiplies the tridiagonal matrix

$\begin{displaymath} A = \left[\begin{array}{rrrr} 1 & 6 & 0 & 0 \\ 2 & -4 & 3 & 0 \\ 0 & -3 & -1 & 1 \end{array}\right] \end{displaymath}$

with the vector x = (1,-1,2,-2).

>>> from cvxopt.base import matrix
>>> from cvxopt.blas import gbmv
>>> A = matrix([[0., 1., 2.],  [6., -4., -3.],  [3., -1., 0.],  [1., 0., 0.]])
>>> x = matrix([1., -1., 2., -2.])
>>> y = matrix(0., (3,1))
>>> gbmv(A, 3, 1, x, y)
>>> print y
[-5.00e+00]
[ 1.20e+01]
[-1.00e+00]

The following example illustrates the use of tbsv().

>>> from cvxopt.base import matrix
>>> from cvxopt.blas import tbsv
>>> A = matrix([-6., 5., -1., 2.], (1,4))
>>> x = matrix(1.0, (4,1))
>>> tbsv(A, x)  # x := diag(A)^{-1}*x
>>> print x
[-1.67e-01]
[ 2.00e-01]
[-1.00e+00]
[ 5.00e-01]

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3.3 Level 2 BLAS