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rl/Lib/site-packages/mpmath/matrices/__init__.py Normal file
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from . import eigen # to set methods
from . import eigen_symmetric # to set methods

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from ..libmp.backend import xrange
# TODO: should use diagonalization-based algorithms
class MatrixCalculusMethods(object):
def _exp_pade(ctx, a):
"""
Exponential of a matrix using Pade approximants.
See G. H. Golub, C. F. van Loan 'Matrix Computations',
third Ed., page 572
TODO:
- find a good estimate for q
- reduce the number of matrix multiplications to improve
performance
"""
def eps_pade(p):
return ctx.mpf(2)**(3-2*p) * \
ctx.factorial(p)**2/(ctx.factorial(2*p)**2 * (2*p + 1))
q = 4
extraq = 8
while 1:
if eps_pade(q) < ctx.eps:
break
q += 1
q += extraq
j = int(max(1, ctx.mag(ctx.mnorm(a,'inf'))))
extra = q
prec = ctx.prec
ctx.dps += extra + 3
try:
a = a/2**j
na = a.rows
den = ctx.eye(na)
num = ctx.eye(na)
x = ctx.eye(na)
c = ctx.mpf(1)
for k in range(1, q+1):
c *= ctx.mpf(q - k + 1)/((2*q - k + 1) * k)
x = a*x
cx = c*x
num += cx
den += (-1)**k * cx
f = ctx.lu_solve_mat(den, num)
for k in range(j):
f = f*f
finally:
ctx.prec = prec
return f*1
def expm(ctx, A, method='taylor'):
r"""
Computes the matrix exponential of a square matrix `A`, which is defined
by the power series
.. math ::
\exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots
With method='taylor', the matrix exponential is computed
using the Taylor series. With method='pade', Pade approximants
are used instead.
**Examples**
Basic examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> expm(zeros(3))
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
>>> expm(eye(3))
[2.71828182845905 0.0 0.0]
[ 0.0 2.71828182845905 0.0]
[ 0.0 0.0 2.71828182845905]
>>> expm([[1,1,0],[1,0,1],[0,1,0]])
[ 3.86814500615414 2.26812870852145 0.841130841230196]
[ 2.26812870852145 2.44114713886289 1.42699786729125]
[0.841130841230196 1.42699786729125 1.6000162976327]
>>> expm([[1,1,0],[1,0,1],[0,1,0]], method='pade')
[ 3.86814500615414 2.26812870852145 0.841130841230196]
[ 2.26812870852145 2.44114713886289 1.42699786729125]
[0.841130841230196 1.42699786729125 1.6000162976327]
>>> expm([[1+j, 0], [1+j,1]])
[(1.46869393991589 + 2.28735528717884j) 0.0]
[ (1.03776739863568 + 3.536943175722j) (2.71828182845905 + 0.0j)]
Matrices with large entries are allowed::
>>> expm(matrix([[1,2],[2,3]])**25)
[5.65024064048415e+2050488462815550 9.14228140091932e+2050488462815550]
[9.14228140091932e+2050488462815550 1.47925220414035e+2050488462815551]
The identity `\exp(A+B) = \exp(A) \exp(B)` does not hold for
noncommuting matrices::
>>> A = hilbert(3)
>>> B = A + eye(3)
>>> chop(mnorm(A*B - B*A))
0.0
>>> chop(mnorm(expm(A+B) - expm(A)*expm(B)))
0.0
>>> B = A + ones(3)
>>> mnorm(A*B - B*A)
1.8
>>> mnorm(expm(A+B) - expm(A)*expm(B))
42.0927851137247
"""
if method == 'pade':
prec = ctx.prec
try:
A = ctx.matrix(A)
ctx.prec += 2*A.rows
res = ctx._exp_pade(A)
finally:
ctx.prec = prec
return res
A = ctx.matrix(A)
prec = ctx.prec
j = int(max(1, ctx.mag(ctx.mnorm(A,'inf'))))
j += int(0.5*prec**0.5)
try:
ctx.prec += 10 + 2*j
tol = +ctx.eps
A = A/2**j
T = A
Y = A**0 + A
k = 2
while 1:
T *= A * (1/ctx.mpf(k))
if ctx.mnorm(T, 'inf') < tol:
break
Y += T
k += 1
for k in xrange(j):
Y = Y*Y
finally:
ctx.prec = prec
Y *= 1
return Y
def cosm(ctx, A):
r"""
Gives the cosine of a square matrix `A`, defined in analogy
with the matrix exponential.
Examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> cosm(X)
[0.54030230586814 0.0 0.0]
[ 0.0 0.54030230586814 0.0]
[ 0.0 0.0 0.54030230586814]
>>> X = hilbert(3)
>>> cosm(X)
[ 0.424403834569555 -0.316643413047167 -0.221474945949293]
[-0.316643413047167 0.820646708837824 -0.127183694770039]
[-0.221474945949293 -0.127183694770039 0.909236687217541]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> cosm(X)
[(0.833730025131149 - 0.988897705762865j) (1.07485840848393 - 0.17192140544213j)]
[ 0.0 (1.54308063481524 + 0.0j)]
"""
B = 0.5 * (ctx.expm(A*ctx.j) + ctx.expm(A*(-ctx.j)))
if not sum(A.apply(ctx.im).apply(abs)):
B = B.apply(ctx.re)
return B
def sinm(ctx, A):
r"""
Gives the sine of a square matrix `A`, defined in analogy
with the matrix exponential.
Examples::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> sinm(X)
[0.841470984807897 0.0 0.0]
[ 0.0 0.841470984807897 0.0]
[ 0.0 0.0 0.841470984807897]
>>> X = hilbert(3)
>>> sinm(X)
[0.711608512150994 0.339783913247439 0.220742837314741]
[0.339783913247439 0.244113865695532 0.187231271174372]
[0.220742837314741 0.187231271174372 0.155816730769635]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> sinm(X)
[(1.29845758141598 + 0.634963914784736j) (-1.96751511930922 + 0.314700021761367j)]
[ 0.0 (0.0 - 1.1752011936438j)]
"""
B = (-0.5j) * (ctx.expm(A*ctx.j) - ctx.expm(A*(-ctx.j)))
if not sum(A.apply(ctx.im).apply(abs)):
B = B.apply(ctx.re)
return B
def _sqrtm_rot(ctx, A, _may_rotate):
# If the iteration fails to converge, cheat by performing
# a rotation by a complex number
u = ctx.j**0.3
return ctx.sqrtm(u*A, _may_rotate) / ctx.sqrt(u)
def sqrtm(ctx, A, _may_rotate=2):
r"""
Computes a square root of the square matrix `A`, i.e. returns
a matrix `B = A^{1/2}` such that `B^2 = A`. The square root
of a matrix, if it exists, is not unique.
**Examples**
Square roots of some simple matrices::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[1,0], [0,1]])
[1.0 0.0]
[0.0 1.0]
>>> sqrtm([[0,0], [0,0]])
[0.0 0.0]
[0.0 0.0]
>>> sqrtm([[2,0],[0,1]])
[1.4142135623731 0.0]
[ 0.0 1.0]
>>> sqrtm([[1,1],[1,0]])
[ (0.920442065259926 - 0.21728689675164j) (0.568864481005783 + 0.351577584254143j)]
[(0.568864481005783 + 0.351577584254143j) (0.351577584254143 - 0.568864481005783j)]
>>> sqrtm([[1,0],[0,1]])
[1.0 0.0]
[0.0 1.0]
>>> sqrtm([[-1,0],[0,1]])
[(0.0 - 1.0j) 0.0]
[ 0.0 (1.0 + 0.0j)]
>>> sqrtm([[j,0],[0,j]])
[(0.707106781186547 + 0.707106781186547j) 0.0]
[ 0.0 (0.707106781186547 + 0.707106781186547j)]
A square root of a rotation matrix, giving the corresponding
half-angle rotation matrix::
>>> t1 = 0.75
>>> t2 = t1 * 0.5
>>> A1 = matrix([[cos(t1), -sin(t1)], [sin(t1), cos(t1)]])
>>> A2 = matrix([[cos(t2), -sin(t2)], [sin(t2), cos(t2)]])
>>> sqrtm(A1)
[0.930507621912314 -0.366272529086048]
[0.366272529086048 0.930507621912314]
>>> A2
[0.930507621912314 -0.366272529086048]
[0.366272529086048 0.930507621912314]
The identity `(A^2)^{1/2} = A` does not necessarily hold::
>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> sqrtm(A**2)
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> sqrtm(A)**2
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> A = matrix([[-4,1,4],[7,-8,9],[10,2,11]])
>>> sqrtm(A**2)
[ 7.43715112194995 -0.324127569985474 1.8481718827526]
[-0.251549715716942 9.32699765900402 2.48221180985147]
[ 4.11609388833616 0.775751877098258 13.017955697342]
>>> chop(sqrtm(A)**2)
[-4.0 1.0 4.0]
[ 7.0 -8.0 9.0]
[10.0 2.0 11.0]
For some matrices, a square root does not exist::
>>> sqrtm([[0,1], [0,0]])
Traceback (most recent call last):
...
ZeroDivisionError: matrix is numerically singular
Two examples from the documentation for Matlab's ``sqrtm``::
>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[7,10],[15,22]])
[1.56669890360128 1.74077655955698]
[2.61116483933547 4.17786374293675]
>>>
>>> X = matrix(\
... [[5,-4,1,0,0],
... [-4,6,-4,1,0],
... [1,-4,6,-4,1],
... [0,1,-4,6,-4],
... [0,0,1,-4,5]])
>>> Y = matrix(\
... [[2,-1,-0,-0,-0],
... [-1,2,-1,0,-0],
... [0,-1,2,-1,0],
... [-0,0,-1,2,-1],
... [-0,-0,-0,-1,2]])
>>> mnorm(sqrtm(X) - Y)
4.53155328326114e-19
"""
A = ctx.matrix(A)
# Trivial
if A*0 == A:
return A
prec = ctx.prec
if _may_rotate:
d = ctx.det(A)
if abs(ctx.im(d)) < 16*ctx.eps and ctx.re(d) < 0:
return ctx._sqrtm_rot(A, _may_rotate-1)
try:
ctx.prec += 10
tol = ctx.eps * 128
Y = A
Z = I = A**0
k = 0
# Denman-Beavers iteration
while 1:
Yprev = Y
try:
Y, Z = 0.5*(Y+ctx.inverse(Z)), 0.5*(Z+ctx.inverse(Y))
except ZeroDivisionError:
if _may_rotate:
Y = ctx._sqrtm_rot(A, _may_rotate-1)
break
else:
raise
mag1 = ctx.mnorm(Y-Yprev, 'inf')
mag2 = ctx.mnorm(Y, 'inf')
if mag1 <= mag2*tol:
break
if _may_rotate and k > 6 and not mag1 < mag2 * 0.001:
return ctx._sqrtm_rot(A, _may_rotate-1)
k += 1
if k > ctx.prec:
raise ctx.NoConvergence
finally:
ctx.prec = prec
Y *= 1
return Y
def logm(ctx, A):
r"""
Computes a logarithm of the square matrix `A`, i.e. returns
a matrix `B = \log(A)` such that `\exp(B) = A`. The logarithm
of a matrix, if it exists, is not unique.
**Examples**
Logarithms of some simple matrices::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> logm(X)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
>>> logm(2*X)
[0.693147180559945 0.0 0.0]
[ 0.0 0.693147180559945 0.0]
[ 0.0 0.0 0.693147180559945]
>>> logm(expm(X))
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
A logarithm of a complex matrix::
>>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]])
>>> B = logm(X)
>>> nprint(B)
[ (0.808757 + 0.107759j) (2.20752 + 0.202762j) (1.07376 - 0.773874j)]
[ (0.905709 - 0.107795j) (0.0287395 - 0.824993j) (0.111619 + 0.514272j)]
[(-0.930151 + 0.399512j) (-2.06266 - 0.674397j) (0.791552 + 0.519839j)]
>>> chop(expm(B))
[(2.0 + 1.0j) 1.0 3.0]
[(1.0 - 1.0j) (1.0 - 2.0j) 1.0]
[ -4.0 -5.0 (0.0 + 1.0j)]
A matrix `X` close to the identity matrix, for which
`\log(\exp(X)) = \exp(\log(X)) = X` holds::
>>> X = eye(3) + hilbert(3)/4
>>> X
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
>>> logm(expm(X))
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
>>> expm(logm(X))
[ 1.25 0.125 0.0833333333333333]
[ 0.125 1.08333333333333 0.0625]
[0.0833333333333333 0.0625 1.05]
A logarithm of a rotation matrix, giving back the angle of
the rotation::
>>> t = 3.7
>>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
>>> chop(logm(A))
[ 0.0 -2.58318530717959]
[2.58318530717959 0.0]
>>> (2*pi-t)
2.58318530717959
For some matrices, a logarithm does not exist::
>>> logm([[1,0], [0,0]])
Traceback (most recent call last):
...
ZeroDivisionError: matrix is numerically singular
Logarithm of a matrix with large entries::
>>> logm(hilbert(3) * 10**20).apply(re)
[ 45.5597513593433 1.27721006042799 0.317662687717978]
[ 1.27721006042799 42.5222778973542 2.24003708791604]
[0.317662687717978 2.24003708791604 42.395212822267]
"""
A = ctx.matrix(A)
prec = ctx.prec
try:
ctx.prec += 10
tol = ctx.eps * 128
I = A**0
B = A
n = 0
while 1:
B = ctx.sqrtm(B)
n += 1
if ctx.mnorm(B-I, 'inf') < 0.125:
break
T = X = B-I
L = X*0
k = 1
while 1:
if k & 1:
L += T / k
else:
L -= T / k
T *= X
if ctx.mnorm(T, 'inf') < tol:
break
k += 1
if k > ctx.prec:
raise ctx.NoConvergence
finally:
ctx.prec = prec
L *= 2**n
return L
def powm(ctx, A, r):
r"""
Computes `A^r = \exp(A \log r)` for a matrix `A` and complex
number `r`.
**Examples**
Powers and inverse powers of a matrix::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> powm(A, 2)
[ 63.0 20.0 69.0]
[174.0 89.0 199.0]
[164.0 48.0 179.0]
>>> chop(powm(powm(A, 4), 1/4.))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> powm(extraprec(20)(powm)(A, -4), -1/4.)
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> chop(powm(powm(A, 1+0.5j), 1/(1+0.5j)))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
>>> powm(extraprec(5)(powm)(A, -1.5), -1/(1.5))
[ 4.0 1.0 4.0]
[ 7.0 8.0 9.0]
[10.0 2.0 11.0]
A Fibonacci-generating matrix::
>>> powm([[1,1],[1,0]], 10)
[89.0 55.0]
[55.0 34.0]
>>> fib(10)
55.0
>>> powm([[1,1],[1,0]], 6.5)
[(16.5166626964253 - 0.0121089837381789j) (10.2078589271083 + 0.0195927472575932j)]
[(10.2078589271083 + 0.0195927472575932j) (6.30880376931698 - 0.0317017309957721j)]
>>> (phi**6.5 - (1-phi)**6.5)/sqrt(5)
(10.2078589271083 - 0.0195927472575932j)
>>> powm([[1,1],[1,0]], 6.2)
[ (14.3076953002666 - 0.008222855781077j) (8.81733464837593 + 0.0133048601383712j)]
[(8.81733464837593 + 0.0133048601383712j) (5.49036065189071 - 0.0215277159194482j)]
>>> (phi**6.2 - (1-phi)**6.2)/sqrt(5)
(8.81733464837593 - 0.0133048601383712j)
"""
A = ctx.matrix(A)
r = ctx.convert(r)
prec = ctx.prec
try:
ctx.prec += 10
if ctx.isint(r):
v = A ** int(r)
elif ctx.isint(r*2):
y = int(r*2)
v = ctx.sqrtm(A) ** y
else:
v = ctx.expm(r*ctx.logm(A))
finally:
ctx.prec = prec
v *= 1
return v

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#!/usr/bin/python
# -*- coding: utf-8 -*-
##################################################################################################
# module for the eigenvalue problem
# Copyright 2013 Timo Hartmann (thartmann15 at gmail.com)
#
# todo:
# - implement balancing
# - agressive early deflation
#
##################################################################################################
"""
The eigenvalue problem
----------------------
This file contains routines for the eigenvalue problem.
high level routines:
hessenberg : reduction of a real or complex square matrix to upper Hessenberg form
schur : reduction of a real or complex square matrix to upper Schur form
eig : eigenvalues and eigenvectors of a real or complex square matrix
low level routines:
hessenberg_reduce_0 : reduction of a real or complex square matrix to upper Hessenberg form
hessenberg_reduce_1 : auxiliary routine to hessenberg_reduce_0
qr_step : a single implicitly shifted QR step for an upper Hessenberg matrix
hessenberg_qr : Schur decomposition of an upper Hessenberg matrix
eig_tr_r : right eigenvectors of an upper triangular matrix
eig_tr_l : left eigenvectors of an upper triangular matrix
"""
from ..libmp.backend import xrange
class Eigen(object):
pass
def defun(f):
setattr(Eigen, f.__name__, f)
return f
def hessenberg_reduce_0(ctx, A, T):
"""
This routine computes the (upper) Hessenberg decomposition of a square matrix A.
Given A, an unitary matrix Q is calculated such that
Q' A Q = H and Q' Q = Q Q' = 1
where H is an upper Hessenberg matrix, meaning that it only contains zeros
below the first subdiagonal. Here ' denotes the hermitian transpose (i.e.
transposition and conjugation).
parameters:
A (input/output) On input, A contains the square matrix A of
dimension (n,n). On output, A contains a compressed representation
of Q and H.
T (output) An array of length n containing the first elements of
the Householder reflectors.
"""
# internally we work with householder reflections from the right.
# let u be a row vector (i.e. u[i]=A[i,:i]). then
# Q is build up by reflectors of the type (1-v'v) where v is a suitable
# modification of u. these reflectors are applyed to A from the right.
# because we work with reflectors from the right we have to start with
# the bottom row of A and work then upwards (this corresponds to
# some kind of RQ decomposition).
# the first part of the vectors v (i.e. A[i,:(i-1)]) are stored as row vectors
# in the lower left part of A (excluding the diagonal and subdiagonal).
# the last entry of v is stored in T.
# the upper right part of A (including diagonal and subdiagonal) becomes H.
n = A.rows
if n <= 2: return
for i in xrange(n-1, 1, -1):
# scale the vector
scale = 0
for k in xrange(0, i):
scale += abs(ctx.re(A[i,k])) + abs(ctx.im(A[i,k]))
scale_inv = 0
if scale != 0:
scale_inv = 1 / scale
if scale == 0 or ctx.isinf(scale_inv):
# sadly there are floating point numbers not equal to zero whose reciprocal is infinity
T[i] = 0
A[i,i-1] = 0
continue
# calculate parameters for housholder transformation
H = 0
for k in xrange(0, i):
A[i,k] *= scale_inv
rr = ctx.re(A[i,k])
ii = ctx.im(A[i,k])
H += rr * rr + ii * ii
F = A[i,i-1]
f = abs(F)
G = ctx.sqrt(H)
A[i,i-1] = - G * scale
if f == 0:
T[i] = G
else:
ff = F / f
T[i] = F + G * ff
A[i,i-1] *= ff
H += G * f
H = 1 / ctx.sqrt(H)
T[i] *= H
for k in xrange(0, i - 1):
A[i,k] *= H
for j in xrange(0, i):
# apply housholder transformation (from right)
G = ctx.conj(T[i]) * A[j,i-1]
for k in xrange(0, i-1):
G += ctx.conj(A[i,k]) * A[j,k]
A[j,i-1] -= G * T[i]
for k in xrange(0, i-1):
A[j,k] -= G * A[i,k]
for j in xrange(0, n):
# apply housholder transformation (from left)
G = T[i] * A[i-1,j]
for k in xrange(0, i-1):
G += A[i,k] * A[k,j]
A[i-1,j] -= G * ctx.conj(T[i])
for k in xrange(0, i-1):
A[k,j] -= G * ctx.conj(A[i,k])
def hessenberg_reduce_1(ctx, A, T):
"""
This routine forms the unitary matrix Q described in hessenberg_reduce_0.
parameters:
A (input/output) On input, A is the same matrix as delivered by
hessenberg_reduce_0. On output, A is set to Q.
T (input) On input, T is the same array as delivered by hessenberg_reduce_0.
"""
n = A.rows
if n == 1:
A[0,0] = 1
return
A[0,0] = A[1,1] = 1
A[0,1] = A[1,0] = 0
for i in xrange(2, n):
if T[i] != 0:
for j in xrange(0, i):
G = T[i] * A[i-1,j]
for k in xrange(0, i-1):
G += A[i,k] * A[k,j]
A[i-1,j] -= G * ctx.conj(T[i])
for k in xrange(0, i-1):
A[k,j] -= G * ctx.conj(A[i,k])
A[i,i] = 1
for j in xrange(0, i):
A[j,i] = A[i,j] = 0
@defun
def hessenberg(ctx, A, overwrite_a = False):
"""
This routine computes the Hessenberg decomposition of a square matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = H and Q' Q = Q Q' = 1
where H is an upper right Hessenberg matrix. Here ' denotes the hermitian
transpose (i.e. transposition and conjugation).
input:
A : a real or complex square matrix
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
Q : an unitary matrix
H : an upper right Hessenberg matrix
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> Q, H = mp.hessenberg(A)
>>> mp.nprint(H, 3) # doctest:+SKIP
[ 3.15 2.23 4.44]
[-0.769 4.85 3.05]
[ 0.0 3.61 7.0]
>>> print(mp.chop(A - Q * H * Q.transpose_conj()))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
return value: (Q, H)
"""
n = A.rows
if n == 1:
return (ctx.matrix([[1]]), A)
if not overwrite_a:
A = A.copy()
T = ctx.matrix(n, 1)
hessenberg_reduce_0(ctx, A, T)
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
for x in xrange(n):
for y in xrange(x+2, n):
A[y,x] = 0
return Q, A
###########################################################################
def qr_step(ctx, n0, n1, A, Q, shift):
"""
This subroutine executes a single implicitly shifted QR step applied to an
upper Hessenberg matrix A. Given A and shift as input, first an QR
decomposition is calculated:
Q R = A - shift * 1 .
The output is then following matrix:
R Q + shift * 1
parameters:
n0, n1 (input) Two integers which specify the submatrix A[n0:n1,n0:n1]
on which this subroutine operators. The subdiagonal elements
to the left and below this submatrix must be deflated (i.e. zero).
following restriction is imposed: n1>=n0+2
A (input/output) On input, A is an upper Hessenberg matrix.
On output, A is replaced by "R Q + shift * 1"
Q (input/output) The parameter Q is multiplied by the unitary matrix
Q arising from the QR decomposition. Q can also be false, in which
case the unitary matrix Q is not computated.
shift (input) a complex number specifying the shift. idealy close to an
eigenvalue of the bottemmost part of the submatrix A[n0:n1,n0:n1].
references:
Stoer, Bulirsch - Introduction to Numerical Analysis.
Kresser : Numerical Methods for General and Structured Eigenvalue Problems
"""
# implicitly shifted and bulge chasing is explained at p.398/399 in "Stoer, Bulirsch - Introduction to Numerical Analysis"
# for bulge chasing see also "Watkins - The Matrix Eigenvalue Problem" sec.4.5,p.173
# the Givens rotation we used is determined as follows: let c,s be two complex
# numbers. then we have following relation:
#
# v = sqrt(|c|^2 + |s|^2)
#
# 1/v [ c~ s~] [c] = [v]
# [-s c ] [s] [0]
#
# the matrix on the left is our Givens rotation.
n = A.rows
# first step
# calculate givens rotation
c = A[n0 ,n0] - shift
s = A[n0+1,n0]
v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s)))
if v == 0:
v = 1
c = 1
s = 0
else:
c /= v
s /= v
cc = ctx.conj(c)
cs = ctx.conj(s)
for k in xrange(n0, n):
# apply givens rotation from the left
x = A[n0 ,k]
y = A[n0+1,k]
A[n0 ,k] = cc * x + cs * y
A[n0+1,k] = c * y - s * x
for k in xrange(min(n1, n0+3)):
# apply givens rotation from the right
x = A[k,n0 ]
y = A[k,n0+1]
A[k,n0 ] = c * x + s * y
A[k,n0+1] = cc * y - cs * x
if not isinstance(Q, bool):
for k in xrange(n):
# eigenvectors
x = Q[k,n0 ]
y = Q[k,n0+1]
Q[k,n0 ] = c * x + s * y
Q[k,n0+1] = cc * y - cs * x
# chase the bulge
for j in xrange(n0, n1 - 2):
# calculate givens rotation
c = A[j+1,j]
s = A[j+2,j]
v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s)))
if v == 0:
A[j+1,j] = 0
v = 1
c = 1
s = 0
else:
A[j+1,j] = v
c /= v
s /= v
A[j+2,j] = 0
cc = ctx.conj(c)
cs = ctx.conj(s)
for k in xrange(j+1, n):
# apply givens rotation from the left
x = A[j+1,k]
y = A[j+2,k]
A[j+1,k] = cc * x + cs * y
A[j+2,k] = c * y - s * x
for k in xrange(0, min(n1, j+4)):
# apply givens rotation from the right
x = A[k,j+1]
y = A[k,j+2]
A[k,j+1] = c * x + s * y
A[k,j+2] = cc * y - cs * x
if not isinstance(Q, bool):
for k in xrange(0, n):
# eigenvectors
x = Q[k,j+1]
y = Q[k,j+2]
Q[k,j+1] = c * x + s * y
Q[k,j+2] = cc * y - cs * x
def hessenberg_qr(ctx, A, Q):
"""
This routine computes the Schur decomposition of an upper Hessenberg matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = R and Q' Q = Q Q' = 1
where R is an upper right triangular matrix. Here ' denotes the hermitian
transpose (i.e. transposition and conjugation).
parameters:
A (input/output) On input, A contains an upper Hessenberg matrix.
On output, A is replace by the upper right triangluar matrix R.
Q (input/output) The parameter Q is multiplied by the unitary
matrix Q arising from the Schur decomposition. Q can also be
false, in which case the unitary matrix Q is not computated.
"""
n = A.rows
norm = 0
for x in xrange(n):
for y in xrange(min(x+2, n)):
norm += ctx.re(A[y,x]) ** 2 + ctx.im(A[y,x]) ** 2
norm = ctx.sqrt(norm) / n
if norm == 0:
return
n0 = 0
n1 = n
eps = ctx.eps / (100 * n)
maxits = ctx.dps * 4
its = totalits = 0
while 1:
# kressner p.32 algo 3
# the active submatrix is A[n0:n1,n0:n1]
k = n0
while k + 1 < n1:
s = abs(ctx.re(A[k,k])) + abs(ctx.im(A[k,k])) + abs(ctx.re(A[k+1,k+1])) + abs(ctx.im(A[k+1,k+1]))
if s < eps * norm:
s = norm
if abs(A[k+1,k]) < eps * s:
break
k += 1
if k + 1 < n1:
# deflation found at position (k+1, k)
A[k+1,k] = 0
n0 = k + 1
its = 0
if n0 + 1 >= n1:
# block of size at most two has converged
n0 = 0
n1 = k + 1
if n1 < 2:
# QR algorithm has converged
return
else:
if (its % 30) == 10:
# exceptional shift
shift = A[n1-1,n1-2]
elif (its % 30) == 20:
# exceptional shift
shift = abs(A[n1-1,n1-2])
elif (its % 30) == 29:
# exceptional shift
shift = norm
else:
# A = [ a b ] det(x-A)=x*x-x*tr(A)+det(A)
# [ c d ]
#
# eigenvalues bad: (tr(A)+sqrt((tr(A))**2-4*det(A)))/2
# bad because of cancellation if |c| is small and |a-d| is small, too.
#
# eigenvalues good: (a+d+sqrt((a-d)**2+4*b*c))/2
t = A[n1-2,n1-2] + A[n1-1,n1-1]
s = (A[n1-1,n1-1] - A[n1-2,n1-2]) ** 2 + 4 * A[n1-1,n1-2] * A[n1-2,n1-1]
if ctx.re(s) > 0:
s = ctx.sqrt(s)
else:
s = ctx.sqrt(-s) * 1j
a = (t + s) / 2
b = (t - s) / 2
if abs(A[n1-1,n1-1] - a) > abs(A[n1-1,n1-1] - b):
shift = b
else:
shift = a
its += 1
totalits += 1
qr_step(ctx, n0, n1, A, Q, shift)
if its > maxits:
raise RuntimeError("qr: failed to converge after %d steps" % its)
@defun
def schur(ctx, A, overwrite_a = False):
"""
This routine computes the Schur decomposition of a square matrix A.
Given A, an unitary matrix Q is determined such that
Q' A Q = R and Q' Q = Q Q' = 1
where R is an upper right triangular matrix. Here ' denotes the
hermitian transpose (i.e. transposition and conjugation).
input:
A : a real or complex square matrix
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
Q : an unitary matrix
R : an upper right triangular matrix
return value: (Q, R)
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> Q, R = mp.schur(A)
>>> mp.nprint(R, 3) # doctest:+SKIP
[2.0 0.417 -2.53]
[0.0 4.0 -4.74]
[0.0 0.0 9.0]
>>> print(mp.chop(A - Q * R * Q.transpose_conj()))
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
warning: The Schur decomposition is not unique.
"""
n = A.rows
if n == 1:
return (ctx.matrix([[1]]), A)
if not overwrite_a:
A = A.copy()
T = ctx.matrix(n, 1)
hessenberg_reduce_0(ctx, A, T)
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
for x in xrange(n):
for y in xrange(x + 2, n):
A[y,x] = 0
hessenberg_qr(ctx, A, Q)
return Q, A
def eig_tr_r(ctx, A):
"""
This routine calculates the right eigenvectors of an upper right triangular matrix.
input:
A an upper right triangular matrix
output:
ER a matrix whose columns form the right eigenvectors of A
return value: ER
"""
# this subroutine is inspired by the lapack routines ctrevc.f,clatrs.f
n = A.rows
ER = ctx.eye(n)
eps = ctx.eps
unfl = ctx.ldexp(ctx.one, -ctx.prec * 30)
# since mpmath effectively has no limits on the exponent, we simply scale doubles up
# original double has prec*20
smlnum = unfl * (n / eps)
simin = 1 / ctx.sqrt(eps)
rmax = 1
for i in xrange(1, n):
s = A[i,i]
smin = max(eps * abs(s), smlnum)
for j in xrange(i - 1, -1, -1):
r = 0
for k in xrange(j + 1, i + 1):
r += A[j,k] * ER[k,i]
t = A[j,j] - s
if abs(t) < smin:
t = smin
r = -r / t
ER[j,i] = r
rmax = max(rmax, abs(r))
if rmax > simin:
for k in xrange(j, i+1):
ER[k,i] /= rmax
rmax = 1
if rmax != 1:
for k in xrange(0, i + 1):
ER[k,i] /= rmax
return ER
def eig_tr_l(ctx, A):
"""
This routine calculates the left eigenvectors of an upper right triangular matrix.
input:
A an upper right triangular matrix
output:
EL a matrix whose rows form the left eigenvectors of A
return value: EL
"""
n = A.rows
EL = ctx.eye(n)
eps = ctx.eps
unfl = ctx.ldexp(ctx.one, -ctx.prec * 30)
# since mpmath effectively has no limits on the exponent, we simply scale doubles up
# original double has prec*20
smlnum = unfl * (n / eps)
simin = 1 / ctx.sqrt(eps)
rmax = 1
for i in xrange(0, n - 1):
s = A[i,i]
smin = max(eps * abs(s), smlnum)
for j in xrange(i + 1, n):
r = 0
for k in xrange(i, j):
r += EL[i,k] * A[k,j]
t = A[j,j] - s
if abs(t) < smin:
t = smin
r = -r / t
EL[i,j] = r
rmax = max(rmax, abs(r))
if rmax > simin:
for k in xrange(i, j + 1):
EL[i,k] /= rmax
rmax = 1
if rmax != 1:
for k in xrange(i, n):
EL[i,k] /= rmax
return EL
@defun
def eig(ctx, A, left = False, right = True, overwrite_a = False):
"""
This routine computes the eigenvalues and optionally the left and right
eigenvectors of a square matrix A. Given A, a vector E and matrices ER
and EL are calculated such that
A ER[:,i] = E[i] ER[:,i]
EL[i,:] A = EL[i,:] E[i]
E contains the eigenvalues of A. The columns of ER contain the right eigenvectors
of A whereas the rows of EL contain the left eigenvectors.
input:
A : a real or complex square matrix of shape (n, n)
left : if true, the left eigenvectors are calculated.
right : if true, the right eigenvectors are calculated.
overwrite_a : if true, allows modification of A which may improve
performance. if false, A is not modified.
output:
E : a list of length n containing the eigenvalues of A.
ER : a matrix whose columns contain the right eigenvectors of A.
EL : a matrix whose rows contain the left eigenvectors of A.
return values:
E if left and right are both false.
(E, ER) if right is true and left is false.
(E, EL) if left is true and right is false.
(E, EL, ER) if left and right are true.
examples:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> E, ER = mp.eig(A)
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> E, EL, ER = mp.eig(A,left = True, right = True)
>>> E, EL, ER = mp.eig_sort(E, EL, ER)
>>> mp.nprint(E)
[2.0, 4.0, 9.0]
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0]))
[0.0 0.0 0.0]
warning:
- If there are multiple eigenvalues, the eigenvectors do not necessarily
span the whole vectorspace, i.e. ER and EL may have not full rank.
Furthermore in that case the eigenvectors are numerical ill-conditioned.
- In the general case the eigenvalues have no natural order.
see also:
- eigh (or eigsy, eighe) for the symmetric eigenvalue problem.
- eig_sort for sorting of eigenvalues and eigenvectors
"""
n = A.rows
if n == 1:
if left and (not right):
return ([A[0]], ctx.matrix([[1]]))
if right and (not left):
return ([A[0]], ctx.matrix([[1]]))
return ([A[0]], ctx.matrix([[1]]), ctx.matrix([[1]]))
if not overwrite_a:
A = A.copy()
T = ctx.zeros(n, 1)
hessenberg_reduce_0(ctx, A, T)
if left or right:
Q = A.copy()
hessenberg_reduce_1(ctx, Q, T)
else:
Q = False
for x in xrange(n):
for y in xrange(x + 2, n):
A[y,x] = 0
hessenberg_qr(ctx, A, Q)
E = [0 for i in xrange(n)]
for i in xrange(n):
E[i] = A[i,i]
if not (left or right):
return E
if left:
EL = eig_tr_l(ctx, A)
EL = EL * Q.transpose_conj()
if right:
ER = eig_tr_r(ctx, A)
ER = Q * ER
if left and (not right):
return (E, EL)
if right and (not left):
return (E, ER)
return (E, EL, ER)
@defun
def eig_sort(ctx, E, EL = False, ER = False, f = "real"):
"""
This routine sorts the eigenvalues and eigenvectors delivered by ``eig``.
parameters:
E : the eigenvalues as delivered by eig
EL : the left eigenvectors as delivered by eig, or false
ER : the right eigenvectors as delivered by eig, or false
f : either a string ("real" sort by increasing real part, "imag" sort by
increasing imag part, "abs" sort by absolute value) or a function
mapping complexs to the reals, i.e. ``f = lambda x: -mp.re(x) ``
would sort the eigenvalues by decreasing real part.
return values:
E if EL and ER are both false.
(E, ER) if ER is not false and left is false.
(E, EL) if EL is not false and right is false.
(E, EL, ER) if EL and ER are not false.
example:
>>> from mpmath import mp
>>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]])
>>> E, EL, ER = mp.eig(A,left = True, right = True)
>>> E, EL, ER = mp.eig_sort(E, EL, ER)
>>> mp.nprint(E)
[2.0, 4.0, 9.0]
>>> E, EL, ER = mp.eig_sort(E, EL, ER,f = lambda x: -mp.re(x))
>>> mp.nprint(E)
[9.0, 4.0, 2.0]
>>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0]))
[0.0]
[0.0]
[0.0]
>>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0]))
[0.0 0.0 0.0]
"""
if isinstance(f, str):
if f == "real":
f = ctx.re
elif f == "imag":
f = ctx.im
elif f == "abs":
f = abs
else:
raise RuntimeError("unknown function %s" % f)
n = len(E)
# Sort eigenvalues (bubble-sort)
for i in xrange(n):
imax = i
s = f(E[i]) # s is the current maximal element
for j in xrange(i + 1, n):
c = f(E[j])
if c < s:
s = c
imax = j
if imax != i:
# swap eigenvalues
z = E[i]
E[i] = E[imax]
E[imax] = z
if not isinstance(EL, bool):
for j in xrange(n):
z = EL[i,j]
EL[i,j] = EL[imax,j]
EL[imax,j] = z
if not isinstance(ER, bool):
for j in xrange(n):
z = ER[j,i]
ER[j,i] = ER[j,imax]
ER[j,imax] = z
if isinstance(EL, bool) and isinstance(ER, bool):
return E
if isinstance(EL, bool) and not(isinstance(ER, bool)):
return (E, ER)
if isinstance(ER, bool) and not(isinstance(EL, bool)):
return (E, EL)
return (E, EL, ER)

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"""
Linear algebra
--------------
Linear equations
................
Basic linear algebra is implemented; you can for example solve the linear
equation system::
x + 2*y = -10
3*x + 4*y = 10
using ``lu_solve``::
>>> from mpmath import *
>>> mp.pretty = False
>>> A = matrix([[1, 2], [3, 4]])
>>> b = matrix([-10, 10])
>>> x = lu_solve(A, b)
>>> x
matrix(
[['30.0'],
['-20.0']])
If you don't trust the result, use ``residual`` to calculate the residual ||A*x-b||::
>>> residual(A, x, b)
matrix(
[['3.46944695195361e-18'],
['3.46944695195361e-18']])
>>> str(eps)
'2.22044604925031e-16'
As you can see, the solution is quite accurate. The error is caused by the
inaccuracy of the internal floating point arithmetic. Though, it's even smaller
than the current machine epsilon, which basically means you can trust the
result.
If you need more speed, use NumPy, or ``fp.lu_solve`` for a floating-point computation.
>>> fp.lu_solve(A, b) # doctest: +ELLIPSIS
matrix(...)
``lu_solve`` accepts overdetermined systems. It is usually not possible to solve
such systems, so the residual is minimized instead. Internally this is done
using Cholesky decomposition to compute a least squares approximation. This means
that that ``lu_solve`` will square the errors. If you can't afford this, use
``qr_solve`` instead. It is twice as slow but more accurate, and it calculates
the residual automatically.
Matrix factorization
....................
The function ``lu`` computes an explicit LU factorization of a matrix::
>>> P, L, U = lu(matrix([[0,2,3],[4,5,6],[7,8,9]]))
>>> print(P)
[0.0 0.0 1.0]
[1.0 0.0 0.0]
[0.0 1.0 0.0]
>>> print(L)
[ 1.0 0.0 0.0]
[ 0.0 1.0 0.0]
[0.571428571428571 0.214285714285714 1.0]
>>> print(U)
[7.0 8.0 9.0]
[0.0 2.0 3.0]
[0.0 0.0 0.214285714285714]
>>> print(P.T*L*U)
[0.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Interval matrices
-----------------
Matrices may contain interval elements. This allows one to perform
basic linear algebra operations such as matrix multiplication
and equation solving with rigorous error bounds::
>>> a = iv.matrix([['0.1','0.3','1.0'],
... ['7.1','5.5','4.8'],
... ['3.2','4.4','5.6']])
>>>
>>> b = iv.matrix(['4','0.6','0.5'])
>>> c = iv.lu_solve(a, b)
>>> print(c)
[ [5.2582327113062568605927528666, 5.25823271130625686059275702219]]
[[-13.1550493962678375411635581388, -13.1550493962678375411635540152]]
[ [7.42069154774972557628979076189, 7.42069154774972557628979190734]]
>>> print(a*c)
[ [3.99999999999999999999999844904, 4.00000000000000000000000155096]]
[[0.599999999999999999999968898009, 0.600000000000000000000031763736]]
[[0.499999999999999999999979320485, 0.500000000000000000000020679515]]
"""
# TODO:
# *implement high-level qr()
# *test unitvector
# *iterative solving
from copy import copy
from ..libmp.backend import xrange
class LinearAlgebraMethods(object):
def LU_decomp(ctx, A, overwrite=False, use_cache=True):
"""
LU-factorization of a n*n matrix using the Gauss algorithm.
Returns L and U in one matrix and the pivot indices.
Use overwrite to specify whether A will be overwritten with L and U.
"""
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
# get from cache if possible
if use_cache and isinstance(A, ctx.matrix) and A._LU:
return A._LU
if not overwrite:
orig = A
A = A.copy()
tol = ctx.absmin(ctx.mnorm(A,1) * ctx.eps) # each pivot element has to be bigger
n = A.rows
p = [None]*(n - 1)
for j in xrange(n - 1):
# pivoting, choose max(abs(reciprocal row sum)*abs(pivot element))
biggest = 0
for k in xrange(j, n):
s = ctx.fsum([ctx.absmin(A[k,l]) for l in xrange(j, n)])
if ctx.absmin(s) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
current = 1/s * ctx.absmin(A[k,j])
if current > biggest: # TODO: what if equal?
biggest = current
p[j] = k
# swap rows according to p
ctx.swap_row(A, j, p[j])
if ctx.absmin(A[j,j]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# calculate elimination factors and add rows
for i in xrange(j + 1, n):
A[i,j] /= A[j,j]
for k in xrange(j + 1, n):
A[i,k] -= A[i,j]*A[j,k]
if ctx.absmin(A[n - 1,n - 1]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# cache decomposition
if not overwrite and isinstance(orig, ctx.matrix):
orig._LU = (A, p)
return A, p
def L_solve(ctx, L, b, p=None):
"""
Solve the lower part of a LU factorized matrix for y.
"""
if L.rows != L.cols:
raise RuntimeError("need n*n matrix")
n = L.rows
if len(b) != n:
raise ValueError("Value should be equal to n")
b = copy(b)
if p: # swap b according to p
for k in xrange(0, len(p)):
ctx.swap_row(b, k, p[k])
# solve
for i in xrange(1, n):
for j in xrange(i):
b[i] -= L[i,j] * b[j]
return b
def U_solve(ctx, U, y):
"""
Solve the upper part of a LU factorized matrix for x.
"""
if U.rows != U.cols:
raise RuntimeError("need n*n matrix")
n = U.rows
if len(y) != n:
raise ValueError("Value should be equal to n")
x = copy(y)
for i in xrange(n - 1, -1, -1):
for j in xrange(i + 1, n):
x[i] -= U[i,j] * x[j]
x[i] /= U[i,i]
return x
def lu_solve(ctx, A, b, **kwargs):
"""
Ax = b => x
Solve a determined or overdetermined linear equations system.
Fast LU decomposition is used, which is less accurate than QR decomposition
(especially for overdetermined systems), but it's twice as efficient.
Use qr_solve if you want more precision or have to solve a very ill-
conditioned system.
If you specify real=True, it does not check for overdeterminded complex
systems.
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
if A.rows > A.cols:
# use least-squares method if overdetermined
# (this increases errors)
AH = A.H
A = AH * A
b = AH * b
if (kwargs.get('real', False) or
not sum(type(i) is ctx.mpc for i in A)):
# TODO: necessary to check also b?
x = ctx.cholesky_solve(A, b)
else:
x = ctx.lu_solve(A, b)
else:
# LU factorization
A, p = ctx.LU_decomp(A)
b = ctx.L_solve(A, b, p)
x = ctx.U_solve(A, b)
finally:
ctx.prec = prec
return x
def improve_solution(ctx, A, x, b, maxsteps=1):
"""
Improve a solution to a linear equation system iteratively.
This re-uses the LU decomposition and is thus cheap.
Usually 3 up to 4 iterations are giving the maximal improvement.
"""
if A.rows != A.cols:
raise RuntimeError("need n*n matrix") # TODO: really?
for _ in xrange(maxsteps):
r = ctx.residual(A, x, b)
if ctx.norm(r, 2) < 10*ctx.eps:
break
# this uses cached LU decomposition and is thus cheap
dx = ctx.lu_solve(A, -r)
x += dx
return x
def lu(ctx, A):
"""
A -> P, L, U
LU factorisation of a square matrix A. L is the lower, U the upper part.
P is the permutation matrix indicating the row swaps.
P*A = L*U
If you need efficiency, use the low-level method LU_decomp instead, it's
much more memory efficient.
"""
# get factorization
A, p = ctx.LU_decomp(A)
n = A.rows
L = ctx.matrix(n)
U = ctx.matrix(n)
for i in xrange(n):
for j in xrange(n):
if i > j:
L[i,j] = A[i,j]
elif i == j:
L[i,j] = 1
U[i,j] = A[i,j]
else:
U[i,j] = A[i,j]
# calculate permutation matrix
P = ctx.eye(n)
for k in xrange(len(p)):
ctx.swap_row(P, k, p[k])
return P, L, U
def unitvector(ctx, n, i):
"""
Return the i-th n-dimensional unit vector.
"""
assert 0 < i <= n, 'this unit vector does not exist'
return [ctx.zero]*(i-1) + [ctx.one] + [ctx.zero]*(n-i)
def inverse(ctx, A, **kwargs):
"""
Calculate the inverse of a matrix.
If you want to solve an equation system Ax = b, it's recommended to use
solve(A, b) instead, it's about 3 times more efficient.
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A
A = ctx.matrix(A, **kwargs).copy()
n = A.rows
# get LU factorisation
A, p = ctx.LU_decomp(A)
cols = []
# calculate unit vectors and solve corresponding system to get columns
for i in xrange(1, n + 1):
e = ctx.unitvector(n, i)
y = ctx.L_solve(A, e, p)
cols.append(ctx.U_solve(A, y))
# convert columns to matrix
inv = []
for i in xrange(n):
row = []
for j in xrange(n):
row.append(cols[j][i])
inv.append(row)
result = ctx.matrix(inv, **kwargs)
finally:
ctx.prec = prec
return result
def householder(ctx, A):
"""
(A|b) -> H, p, x, res
(A|b) is the coefficient matrix with left hand side of an optionally
overdetermined linear equation system.
H and p contain all information about the transformation matrices.
x is the solution, res the residual.
"""
if not isinstance(A, ctx.matrix):
raise TypeError("A should be a type of ctx.matrix")
m = A.rows
n = A.cols
if m < n - 1:
raise RuntimeError("Columns should not be less than rows")
# calculate Householder matrix
p = []
for j in xrange(0, n - 1):
s = ctx.fsum(abs(A[i,j])**2 for i in xrange(j, m))
if not abs(s) > ctx.eps:
raise ValueError('matrix is numerically singular')
p.append(-ctx.sign(ctx.re(A[j,j])) * ctx.sqrt(s))
kappa = ctx.one / (s - p[j] * A[j,j])
A[j,j] -= p[j]
for k in xrange(j+1, n):
y = ctx.fsum(ctx.conj(A[i,j]) * A[i,k] for i in xrange(j, m)) * kappa
for i in xrange(j, m):
A[i,k] -= A[i,j] * y
# solve Rx = c1
x = [A[i,n - 1] for i in xrange(n - 1)]
for i in xrange(n - 2, -1, -1):
x[i] -= ctx.fsum(A[i,j] * x[j] for j in xrange(i + 1, n - 1))
x[i] /= p[i]
# calculate residual
if not m == n - 1:
r = [A[m-1-i, n-1] for i in xrange(m - n + 1)]
else:
# determined system, residual should be 0
r = [0]*m # maybe a bad idea, changing r[i] will change all elements
return A, p, x, r
#def qr(ctx, A):
# """
# A -> Q, R
#
# QR factorisation of a square matrix A using Householder decomposition.
# Q is orthogonal, this leads to very few numerical errors.
#
# A = Q*R
# """
# H, p, x, res = householder(A)
# TODO: implement this
def residual(ctx, A, x, b, **kwargs):
"""
Calculate the residual of a solution to a linear equation system.
r = A*x - b for A*x = b
"""
oldprec = ctx.prec
try:
ctx.prec *= 2
A, x, b = ctx.matrix(A, **kwargs), ctx.matrix(x, **kwargs), ctx.matrix(b, **kwargs)
return A*x - b
finally:
ctx.prec = oldprec
def qr_solve(ctx, A, b, norm=None, **kwargs):
"""
Ax = b => x, ||Ax - b||
Solve a determined or overdetermined linear equations system and
calculate the norm of the residual (error).
QR decomposition using Householder factorization is applied, which gives very
accurate results even for ill-conditioned matrices. qr_solve is twice as
efficient.
"""
if norm is None:
norm = ctx.norm
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
H, p, x, r = ctx.householder(ctx.extend(A, b))
res = ctx.norm(r)
# calculate residual "manually" for determined systems
if res == 0:
res = ctx.norm(ctx.residual(A, x, b))
return ctx.matrix(x, **kwargs), res
finally:
ctx.prec = prec
def cholesky(ctx, A, tol=None):
r"""
Cholesky decomposition of a symmetric positive-definite matrix `A`.
Returns a lower triangular matrix `L` such that `A = L \times L^T`.
More generally, for a complex Hermitian positive-definite matrix,
a Cholesky decomposition satisfying `A = L \times L^H` is returned.
The Cholesky decomposition can be used to solve linear equation
systems twice as efficiently as LU decomposition, or to
test whether `A` is positive-definite.
The optional parameter ``tol`` determines the tolerance for
verifying positive-definiteness.
**Examples**
Cholesky decomposition of a positive-definite symmetric matrix::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> A = eye(3) + hilbert(3)
>>> nprint(A)
[ 2.0 0.5 0.333333]
[ 0.5 1.33333 0.25]
[0.333333 0.25 1.2]
>>> L = cholesky(A)
>>> nprint(L)
[ 1.41421 0.0 0.0]
[0.353553 1.09924 0.0]
[0.235702 0.15162 1.05899]
>>> chop(A - L*L.T)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
Cholesky decomposition of a Hermitian matrix::
>>> A = eye(3) + matrix([[0,0.25j,-0.5j],[-0.25j,0,0],[0.5j,0,0]])
>>> L = cholesky(A)
>>> nprint(L)
[ 1.0 0.0 0.0]
[(0.0 - 0.25j) (0.968246 + 0.0j) 0.0]
[ (0.0 + 0.5j) (0.129099 + 0.0j) (0.856349 + 0.0j)]
>>> chop(A - L*L.H)
[0.0 0.0 0.0]
[0.0 0.0 0.0]
[0.0 0.0 0.0]
Attempted Cholesky decomposition of a matrix that is not positive
definite::
>>> A = -eye(3) + hilbert(3)
>>> L = cholesky(A)
Traceback (most recent call last):
...
ValueError: matrix is not positive-definite
**References**
1. [Wikipedia]_ http://en.wikipedia.org/wiki/Cholesky_decomposition
"""
if not isinstance(A, ctx.matrix):
raise RuntimeError("A should be a type of ctx.matrix")
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
if tol is None:
tol = +ctx.eps
n = A.rows
L = ctx.matrix(n)
for j in xrange(n):
c = ctx.re(A[j,j])
if abs(c-A[j,j]) > tol:
raise ValueError('matrix is not Hermitian')
s = c - ctx.fsum((L[j,k] for k in xrange(j)),
absolute=True, squared=True)
if s < tol:
raise ValueError('matrix is not positive-definite')
L[j,j] = ctx.sqrt(s)
for i in xrange(j, n):
it1 = (L[i,k] for k in xrange(j))
it2 = (L[j,k] for k in xrange(j))
t = ctx.fdot(it1, it2, conjugate=True)
L[i,j] = (A[i,j] - t) / L[j,j]
return L
def cholesky_solve(ctx, A, b, **kwargs):
"""
Ax = b => x
Solve a symmetric positive-definite linear equation system.
This is twice as efficient as lu_solve.
Typical use cases:
* A.T*A
* Hessian matrix
* differential equations
"""
prec = ctx.prec
try:
ctx.prec += 10
# do not overwrite A nor b
A, b = ctx.matrix(A, **kwargs).copy(), ctx.matrix(b, **kwargs).copy()
if A.rows != A.cols:
raise ValueError('can only solve determined system')
# Cholesky factorization
L = ctx.cholesky(A)
# solve
n = L.rows
if len(b) != n:
raise ValueError("Value should be equal to n")
for i in xrange(n):
b[i] -= ctx.fsum(L[i,j] * b[j] for j in xrange(i))
b[i] /= L[i,i]
x = ctx.U_solve(L.T, b)
return x
finally:
ctx.prec = prec
def det(ctx, A):
"""
Calculate the determinant of a matrix.
"""
prec = ctx.prec
try:
# do not overwrite A
A = ctx.matrix(A).copy()
# use LU factorization to calculate determinant
try:
R, p = ctx.LU_decomp(A)
except ZeroDivisionError:
return 0
z = 1
for i, e in enumerate(p):
if i != e:
z *= -1
for i in xrange(A.rows):
z *= R[i,i]
return z
finally:
ctx.prec = prec
def cond(ctx, A, norm=None):
"""
Calculate the condition number of a matrix using a specified matrix norm.
The condition number estimates the sensitivity of a matrix to errors.
Example: small input errors for ill-conditioned coefficient matrices
alter the solution of the system dramatically.
For ill-conditioned matrices it's recommended to use qr_solve() instead
of lu_solve(). This does not help with input errors however, it just avoids
to add additional errors.
Definition: cond(A) = ||A|| * ||A**-1||
"""
if norm is None:
norm = lambda x: ctx.mnorm(x,1)
return norm(A) * norm(ctx.inverse(A))
def lu_solve_mat(ctx, a, b):
"""Solve a * x = b where a and b are matrices."""
r = ctx.matrix(a.rows, b.cols)
for i in range(b.cols):
c = ctx.lu_solve(a, b.column(i))
for j in range(len(c)):
r[j, i] = c[j]
return r
def qr(ctx, A, mode = 'full', edps = 10):
"""
Compute a QR factorization $A = QR$ where
A is an m x n matrix of real or complex numbers where m >= n
mode has following meanings:
(1) mode = 'raw' returns two matrixes (A, tau) in the
internal format used by LAPACK
(2) mode = 'skinny' returns the leading n columns of Q
and n rows of R
(3) Any other value returns the leading m columns of Q
and m rows of R
edps is the increase in mp precision used for calculations
**Examples**
>>> from mpmath import *
>>> mp.dps = 15
>>> mp.pretty = True
>>> A = matrix([[1, 2], [3, 4], [1, 1]])
>>> Q, R = qr(A)
>>> Q
[-0.301511344577764 0.861640436855329 0.408248290463863]
[-0.904534033733291 -0.123091490979333 -0.408248290463863]
[-0.301511344577764 -0.492365963917331 0.816496580927726]
>>> R
[-3.3166247903554 -4.52267016866645]
[ 0.0 0.738548945875996]
[ 0.0 0.0]
>>> Q * R
[1.0 2.0]
[3.0 4.0]
[1.0 1.0]
>>> chop(Q.T * Q)
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]
>>> B = matrix([[1+0j, 2-3j], [3+j, 4+5j]])
>>> Q, R = qr(B)
>>> nprint(Q)
[ (-0.301511 + 0.0j) (0.0695795 - 0.95092j)]
[(-0.904534 - 0.301511j) (-0.115966 + 0.278318j)]
>>> nprint(R)
[(-3.31662 + 0.0j) (-5.72872 - 2.41209j)]
[ 0.0 (3.91965 + 0.0j)]
>>> Q * R
[(1.0 + 0.0j) (2.0 - 3.0j)]
[(3.0 + 1.0j) (4.0 + 5.0j)]
>>> chop(Q.T * Q.conjugate())
[1.0 0.0]
[0.0 1.0]
"""
# check values before continuing
assert isinstance(A, ctx.matrix)
m = A.rows
n = A.cols
assert n >= 0
assert m >= n
assert edps >= 0
# check for complex data type
cmplx = any(type(x) is ctx.mpc for x in A)
# temporarily increase the precision and initialize
with ctx.extradps(edps):
tau = ctx.matrix(n,1)
A = A.copy()
# ---------------
# FACTOR MATRIX A
# ---------------
if cmplx:
one = ctx.mpc('1.0', '0.0')
zero = ctx.mpc('0.0', '0.0')
rzero = ctx.mpf('0.0')
# main loop to factor A (complex)
for j in xrange(0, n):
alpha = A[j,j]
alphr = ctx.re(alpha)
alphi = ctx.im(alpha)
if (m-j) >= 2:
xnorm = ctx.fsum( A[i,j]*ctx.conj(A[i,j]) for i in xrange(j+1, m) )
xnorm = ctx.re( ctx.sqrt(xnorm) )
else:
xnorm = rzero
if (xnorm == rzero) and (alphi == rzero):
tau[j] = zero
continue
if alphr < rzero:
beta = ctx.sqrt(alphr**2 + alphi**2 + xnorm**2)
else:
beta = -ctx.sqrt(alphr**2 + alphi**2 + xnorm**2)
tau[j] = ctx.mpc( (beta - alphr) / beta, -alphi / beta )
t = -ctx.conj(tau[j])
za = one / (alpha - beta)
for i in xrange(j+1, m):
A[i,j] *= za
A[j,j] = one
for k in xrange(j+1, n):
y = ctx.fsum(A[i,j] * ctx.conj(A[i,k]) for i in xrange(j, m))
temp = t * ctx.conj(y)
for i in xrange(j, m):
A[i,k] += A[i,j] * temp
A[j,j] = ctx.mpc(beta, '0.0')
else:
one = ctx.mpf('1.0')
zero = ctx.mpf('0.0')
# main loop to factor A (real)
for j in xrange(0, n):
alpha = A[j,j]
if (m-j) > 2:
xnorm = ctx.fsum( (A[i,j])**2 for i in xrange(j+1, m) )
xnorm = ctx.sqrt(xnorm)
elif (m-j) == 2:
xnorm = abs( A[m-1,j] )
else:
xnorm = zero
if xnorm == zero:
tau[j] = zero
continue
if alpha < zero:
beta = ctx.sqrt(alpha**2 + xnorm**2)
else:
beta = -ctx.sqrt(alpha**2 + xnorm**2)
tau[j] = (beta - alpha) / beta
t = -tau[j]
da = one / (alpha - beta)
for i in xrange(j+1, m):
A[i,j] *= da
A[j,j] = one
for k in xrange(j+1, n):
y = ctx.fsum( A[i,j] * A[i,k] for i in xrange(j, m) )
temp = t * y
for i in xrange(j,m):
A[i,k] += A[i,j] * temp
A[j,j] = beta
# return factorization in same internal format as LAPACK
if (mode == 'raw') or (mode == 'RAW'):
return A, tau
# ----------------------------------
# FORM Q USING BACKWARD ACCUMULATION
# ----------------------------------
# form R before the values are overwritten
R = A.copy()
for j in xrange(0, n):
for i in xrange(j+1, m):
R[i,j] = zero
# set the value of p (number of columns of Q to return)
p = m
if (mode == 'skinny') or (mode == 'SKINNY'):
p = n
# add columns to A if needed and initialize
A.cols += (p-n)
for j in xrange(0, p):
A[j,j] = one
for i in xrange(0, j):
A[i,j] = zero
# main loop to form Q
for j in xrange(n-1, -1, -1):
t = -tau[j]
A[j,j] += t
for k in xrange(j+1, p):
if cmplx:
y = ctx.fsum(A[i,j] * ctx.conj(A[i,k]) for i in xrange(j+1, m))
temp = t * ctx.conj(y)
else:
y = ctx.fsum(A[i,j] * A[i,k] for i in xrange(j+1, m))
temp = t * y
A[j,k] = temp
for i in xrange(j+1, m):
A[i,k] += A[i,j] * temp
for i in xrange(j+1, m):
A[i, j] *= t
return A, R[0:p,0:n]
# ------------------
# END OF FUNCTION QR
# ------------------

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