scipy tutorial - Baustatik-Info-Server

print U
[[-0.70710678 -0.70710678]
[-0.70710678 0.70710678]]
>>> print Sig
[[ 5.19615242 0. 0. ]
[ 0. 1. 0. ]]
>>> print Vh
[[ -2.72165527e-01 -6.80413817e-01 -6.80413817e-01]
[ -6.18652536e-16 -7.07106781e-01 7.07106781e-01]
[ -9.62250449e-01 1.92450090e-01 1.92450090e-01]]
>>> print A
[[1 3 2]
[1 2 3]]
>>> print U*Sig*Vh
[[ 1. 3. 2.]
[ 1. 2. 3.]]
LU decomposition
The LU decompostion finds a representation for the M × N matrix A as
A = PLU
SciPy Reference Guide, Release 0.8.dev
where P is an M × M permutation matrix (a permutation of the rows of the identity matrix), L is in M × K lower
triangular or trapezoidal matrix ( K = min (M, N) ) with unit-diagonal, and U is an upper triangular or trapezoidal
matrix. The SciPy command for this decomposition is linalg.lu .
Such a decomposition is often useful for solving many simultaneous equations where the left-hand-side does not
change but the right hand side does. For example, suppose we are going to solve
Axi = bi
for many different bi . The LU decomposition allows this to be written as
PLUxi = bi.
Because L is lower-triangular, the equation can be solved for Uxi and finally xi very rapidly using forward- and
back-substitution. An initial time spent factoring A allows for very rapid solution of similar systems of equations
in the future. If the intent for performing LU decomposition is for solving linear systems then the command
linalg.lu_factor should be used followed by repeated applications of the command linalg.lu_solve to
solve the system for each new right-hand-side.
Cholesky decomposition
Cholesky decomposition is a special case of LU decomposition applicable to Hermitian positive definite matrices.
When A = A H and x H Ax ≥ 0 for all x , then decompositions of A can be found so that
A = U H U
A = LL H
where L is lower-triangular and U is upper triangular. Notice that L = U H . The command linagl.cholesky
computes the cholesky factorization. For using cholesky factorization to solve systems of equations there are also
linalg.cho_factor and linalg.cho_solve routines that work similarly to their LU decomposition counterparts.
1.8. Linear Algebra 45