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ENH: Add QZ decomposition with tests #185

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5 changes: 5 additions & 0 deletions doc/release/0.11.0-notes.rst
Expand Up @@ -67,6 +67,11 @@ A function for creating Pascal matrices, ``scipy.linalg.pascal``, was added.


``misc.logsumexp`` now takes an optional ``axis`` keyword argument. ``misc.logsumexp`` now takes an optional ``axis`` keyword argument.


QZ Decomposition
----------------

It is now possible to calculate the QZ, or Generalized Schur, decomposition using `scipy.linalg.qz`. This function wraps the LAPACK routines sgges, dgges, cgges, and zgges.



Deprecated features Deprecated features
=================== ===================
Expand Down
2 changes: 2 additions & 0 deletions scipy/linalg/__init__.py
Expand Up @@ -68,6 +68,7 @@
cho_solve_banded - Solve previously factored banded linear system cho_solve_banded - Solve previously factored banded linear system
qr - QR decomposition of a matrix qr - QR decomposition of a matrix
qr_multiply - QR decomposition and multiplication by Q qr_multiply - QR decomposition and multiplication by Q
qz - QZ decomposition of a pair of matrices
schur - Schur decomposition of a matrix schur - Schur decomposition of a matrix
rsf2csf - Real to complex Schur form rsf2csf - Real to complex Schur form
hessenberg - Hessenberg form of a matrix hessenberg - Hessenberg form of a matrix
Expand Down Expand Up @@ -134,6 +135,7 @@
from decomp_lu import * from decomp_lu import *
from decomp_cholesky import * from decomp_cholesky import *
from decomp_qr import * from decomp_qr import *
from _decomp_qz import *
from decomp_svd import * from decomp_svd import *
from decomp_schur import * from decomp_schur import *
from matfuncs import * from matfuncs import *
Expand Down
194 changes: 194 additions & 0 deletions scipy/linalg/_decomp_qz.py
@@ -0,0 +1,194 @@
import warnings

import numpy as np
from numpy import asarray_chkfinite, single

from misc import LinAlgError, _datacopied
from lapack import get_lapack_funcs

__all__ = ['qz']

_double_precision = ['i','l','d']

def _select_function(sort, typ):
if typ in ['F','D']:
if callable(sort):
#assume the user knows what they're doing
sfunction = sort
elif sort == 'lhp':
sfunction = lambda x,y: (np.real(x/y) < 0.0)
elif sort == 'rhp':
sfunction = lambda x,y: (np.real(x/y) >= 0.0)
elif sort == 'iuc':
sfunction = lambda x,y: (abs(x/y) <= 1.0)
elif sort == 'ouc':
sfunction = lambda x,y: (abs(x/y) > 1.0)
else:
raise ValueError("sort parameter must be None, a callable, or "
"one of ('lhp','rhp','iuc','ouc')")
elif typ in ['f','d']:
if callable(sort):
#assume the user knows what they're doing
sfunction = sort
elif sort == 'lhp':
sfunction = lambda x,y,z: (np.real((x+y*1j)/z) < 0.0)
elif sort == 'rhp':
sfunction = lambda x,y,z: (np.real((x+y*1j)/z) >= 0.0)
elif sort == 'iuc':
sfunction = lambda x,y,z: (abs((x+y*1j)/z) <= 1.0)
elif sort == 'ouc':
sfunction = lambda x,y,z: (abs((x+y*1j)/z) > 1.0)
else:
raise ValueError("sort parameter must be None, a callable, or "
"one of ('lhp','rhp','iuc','ouc')")
else: # to avoid an error later
raise ValueError("dtype %s not understood" % typ)
return sfunction


def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
overwrite_b=False):
"""
QZ decompostion for generalized eigenvalues of a pair of matrices.

The QZ, or generalized Schur, decomposition for a pair of N x N
nonsymmetric matrices (A,B) is

(A,B) = (Q*AA*Z', Q*BB*Z')

where AA, BB is in generalized Schur form if BB is upper-triangular
with non-negative diagonal and AA is upper-triangular, or for real QZ
decomposition (output='real') block upper triangular with 1x1
and 2x2 blocks. In this case, the 1x1 blocks correpsond to real
generalized eigenvalues and 2x2 blocks are 'standardized' by making
the correpsonding elements of BB have the form::

[ a 0 ]
[ 0 b ]

and the pair of correpsonding 2x2 blocks in AA and BB will have a complex
conjugate pair of generalized eigenvalues. If (output='complex') or A
and B are complex matrices, Z' denotes the conjugate-transpose of Z.
Q and Z are unitary matrices.

Parameters
----------
A : array-like, shape (N,N)
2d array to decompose
B : array-like, shape (N,N)
2d array to decompose
output : str {'real','complex'}
Construct the real or complex QZ decomposition for real matrices.
lwork : integer, optional
Work array size. If None or -1, it is automatically computed.
sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}
Specifies whether the upper eigenvalues should be sorted. A callable
may be passed that, given a eigenvalue, returns a boolean denoting
whether the eigenvalue should be sorted to the top-left (True). For
real matrix pairs, the sort function takes three real arguments
(alphar, alphai, beta). The eigenvalue x = (alphar + alphai*1j)/beta.
For complex matrix pairs or output='complex', the sort function
takes two complex arguments (alpha, beta). The eigenvalue
x = (alpha/beta).
Alternatively, string parameters may be used:
'lhp' Left-hand plane (x.real < 0.0)
'rhp' Right-hand plane (x.real > 0.0)
'iuc' Inside the unit circle (x*x.conjugate() <= 1.0)
'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
Defaults to None (no sorting).

Returns
-------
AA : array, shape (N,N)
Generalized Schur form of A.
BB : array, shape (N,N)
Generalized Schur form of B.
Q : array, shape (N,N)
The left Schur vectors.
Z : array, shape (N,N)
The right Schur vectors.
sdim : int
If sorting was requested, a fifth return value will contain the
number of eigenvalues for which the sort condition was True.

Notes
-----
Q is transposed versus the equivalent function in Matlab.

.. versionadded:: 0.11.0
"""
if not output in ['real','complex','r','c']:
raise ValueError("argument must be 'real', or 'complex'")

a1 = asarray_chkfinite(A)
b1 = asarray_chkfinite(B)

a_m, a_n = a1.shape
b_m, b_n = b1.shape
try:
assert a_m == a_n == b_m == b_n
except AssertionError:
raise ValueError("Array dimensions must be square and agree")

typa = a1.dtype.char
if output in ['complex', 'c'] and typa not in ['F','D']:
if typa in _double_precision:
a1 = a1.astype('D')
typa = 'D'
else:
a1 = a1.astype('F')
typa = 'F'
typb = b1.dtype.char
if output in ['complex', 'c'] and typb not in ['F','D']:
if typb in _double_precision:
b1 = b1.astype('D')
typb = 'D'
else:
b1 = b1.astype('F')
typb = 'F'

overwrite_a = overwrite_a or (_datacopied(a1,A))
overwrite_b = overwrite_b or (_datacopied(b1,B))

gges, = get_lapack_funcs(('gges',), (a1,b1))

if lwork is None or lwork == -1:
# get optimal work array size
result = gges(lambda x: None, a1, b1, lwork=-1)
lwork = result[-2][0].real.astype(np.int)

if sort is None:
sort_t = 0
sfunction = lambda x : None
else:
sort_t = 1
sfunction = _select_function(sort, typa)

result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a,
overwrite_b=overwrite_b, sort_t=sort_t)

info = result[-1]
if info < 0:
raise ValueError("Illegal value in argument %d of gges" % -info)
elif info > 0 and info <= a_n:
warnings.warn("The QZ iteration failed. (a,b) are not in Schur "
"form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct"
"for J=%d,...,N" % info-1, UserWarning)
elif info == a_n+1:
raise LinAlgError("Something other than QZ iteration failed")
elif info == a_n+2:
raise LinAlgError("After reordering, roundoff changed values of some"
"complex eigenvalues so that leading eigenvalues in the"
"Generalized Schur form no longer satisfy sort=True."
"This could also be caused due to scaling.")
elif info == a_n+3:
raise LinAlgError("Reordering failed in <s,d,c,z>tgsen")

# output for real
#AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info
# output for complex
#AA, BB, sdim, alphai, beta, vsl, vsr, work, info
if sort_t == 0:
return result[0], result[1], result[-4], result[-3]
else:
return result[0], result[1], result[-4], result[-3], result[2]
73 changes: 73 additions & 0 deletions scipy/linalg/flapack.pyf.src
Expand Up @@ -6,6 +6,7 @@
! $Revision$ $Date$ ! $Revision$ $Date$
! !
! Additions by Travis Oliphant, Tiziano Zito, Collin RM Stocks, Fabian Pedregosa ! Additions by Travis Oliphant, Tiziano Zito, Collin RM Stocks, Fabian Pedregosa
! Skipper Seabold
! !
! <prefix2=s,d> <ctype2=float,double> <ftype2=real,double precision> ! <prefix2=s,d> <ctype2=float,double> <ftype2=real,double precision>
! <prefix2c=c,z> <ftype2c=complex,double complex> <ctype2c=complex_float,complex_double> ! <prefix2c=c,z> <ftype2c=complex,double complex> <ctype2c=complex_float,complex_double>
Expand All @@ -15,6 +16,78 @@ interface


include 'flapack_user.pyf.src' include 'flapack_user.pyf.src'


subroutine <prefix2>gges(jobvsl,jobvsr,sort_t,<prefix2>select,n,a,lda,b,ldb,sdim,alphar,alphai,beta,vsl,ldvsl,vsr,ldvsr,work,lwork,bwork,info)

! For a pair of N-by-N real nonsymmetric matrices (A,B) computes
! the generalized eigenvalues, the generalized real Schur form (S,T),
! optionally, the left and/or right matrices of Schur vectors (VSL
! and VSR).
! (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

callstatement (*f2py_func)((jobvsl?"V":"N"),(jobvsr?"V":"N"),(sort_t?"S":"N"),cb_<prefix2>select_in_<prefix2>gges__user__routines,&n,a,&lda,b,&ldb,&sdim,alphar,alphai,beta,vsl,&ldvsl,vsr,&ldvsr,work,&lwork,bwork,&info)
callprotoargument char*,char*,char*,int(*)(<ctype2>*,<ctype2>*,<ctype2>*),int*,<ctype2>*,int*,<ctype2>*,int*,int*,<ctype2>*,<ctype2>*,<ctype2>*,<ctype2>*,int*,<ctype2>*,int*,<ctype2>*,int*,int*,int*

use <prefix2>gges__user__routines

integer optional,intent(in),check(jobvsl==0||jobvsl==1) :: jobvsl=1
integer optional,intent(in),check(jobvsr==0||jobvsr==1) :: jobvsr=1
integer optional,intent(in),check(sort_t==0||sort_t==1) :: sort_t=0
external <prefix2>select
integer intent(hide),depend(a) :: n=shape(a,1)
<ftype2> intent(in,out,copy),dimension(lda,*) :: a
integer intent(hide),depend(a) :: lda=shape(a,0)
<ftype2> intent(in,out,copy),dimension(ldb,*) :: b
integer intent(hide),depend(b) :: ldb=shape(b,0)
integer intent(out) :: sdim=0
<ftype2> intent(out),dimension(n) :: alphar
<ftype2> intent(out),dimension(n) :: alphai
<ftype2> intent(out),dimension(n) :: beta
<ftype2> intent(out),depend(ldvsl,n),dimension(ldvsl,n) :: vsl
integer optional,intent(in),depend(jobvsl,n) :: ldvsl=((jobvsl==1)?n:1)
<ftype2> intent(out),depend(ldvsr,n),dimension(ldvsr,n) :: vsr
integer optional,intent(in),depend(jobvsr,n) :: ldvsr=((jobvsr==1)?n:1)
<ftype2> intent(out),depend(lwork),dimension(MAX(lwork,1)) :: work
integer optional,intent(in),depend(n),check(lwork>=MAX(1,MAX(8*n, 6*n+16))||lwork==-1):: lwork=8*n+16
logical optional,intent(hide),depend(n),dimension(n) :: bwork
integer intent(out):: info
end subroutine <prefix2>gges

subroutine <prefix2c>gges(jobvsl,jobvsr,sort_t,<prefix2c>select,n,a,lda,b,ldb,sdim,alpha,beta,vsl,ldvsl,vsr,ldvsr,work,lwork,rwork,bwork,info)

! For a pair of N-by-N complex nonsymmetric matrices (A,B) computes
! the generalized eigenvalues, the generalized real Schur form (S,T),
! optionally, the left and/or right matrices of Schur vectors (VSL
! and VSR).
! (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**H )

callstatement (*f2py_func)((jobvsl?"V":"N"),(jobvsr?"V":"N"),(sort_t?"S":"N"),cb_<prefix2c>select_in_<prefix2c>gges__user__routines,&n,a,&lda,b,&ldb,&sdim,alpha,beta,vsl,&ldvsl,vsr,&ldvsr,work,&lwork,rwork,bwork,&info)
callprotoargument char*,char*,char*,int(*)(<ctype2c>*,<ctype2c>*),int*,<ctype2c>*,int*,<ctype2c>*,int*,int*,<ctype2c>*,<ctype2c>*,<ctype2c>*,int*,<ctype2c>*,int*,<ctype2c>*,int*,<ctype2>*,int*,int*

use <prefix2c>gges__user__routines

integer optional,intent(in),check(jobvsl==0||jobvsl==1) :: jobvsl=1
integer optional,intent(in),check(jobvsr==0||jobvsr==1) :: jobvsr=1
integer optional,intent(in),check(sort_t==0||sort_t==1) :: sort_t=0
external <prefix2c>select
integer intent(hide),depend(a) :: n=shape(a,1)
<ftype2c> intent(in,out,copy),dimension(lda,*) :: a
integer intent(hide),depend(a) :: lda=shape(a,0)
<ftype2c> intent(in,out,copy),dimension(ldb,*) :: b
integer intent(hide),depend(b) :: ldb=shape(b,0)
integer intent(out) :: sdim=0
<ftype2c> intent(out),dimension(n) :: alpha
<ftype2c> intent(out),dimension(n) :: beta
<ftype2c> intent(out),depend(ldvsl,n),dimension(ldvsl,n) :: vsl
integer optional,intent(in),depend(jobvsl,n) :: ldvsl=((jobvsl==1)?n:1)
<ftype2c> intent(out),depend(ldvsr,n),dimension(ldvsr,n) :: vsr
integer optional,intent(in),depend(jobvsr,n) :: ldvsr=((jobvsr==1)?n:1)
<ftype2c> intent(out),depend(lwork),dimension(MAX(lwork,1)) :: work
integer optional,intent(in),depend(n),check(lwork>=MAX(1,2*n)||lwork==-1):: lwork=2*n
<ftype2> intent(hide),dimension(8*n) :: rwork
logical optional,intent(hide),depend(n),dimension(n) :: bwork
integer intent(out):: info
end subroutine <prefix2c>gges

subroutine <prefix2>pbtrf(lower,n,kd,ab,ldab,info) subroutine <prefix2>pbtrf(lower,n,kd,ab,ldab,info)


! Compute Cholesky decomposition of banded symmetric positive definite ! Compute Cholesky decomposition of banded symmetric positive definite
Expand Down
42 changes: 42 additions & 0 deletions scipy/linalg/flapack_user.pyf.src
Expand Up @@ -6,3 +6,45 @@ python module gees__user__routines
end function <prefix>select end function <prefix>select
end interface end interface
end python module gees__user__routines end python module gees__user__routines

python module zgges__user__routines
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Here you could use f2py's templating:

The syntax <_arg=arg1,arg2,\0,arg,\2> means the same as <_arg=arg1,arg2,arg1,arg2,arg,arg> -- one of the slashes escapes the comma, and apparently the syntax \0, \1, \2, ... can be used (but IMO shouldn't be because it's confusing) to refer to other values in the expansion.

< ftype> :: <_arg>

expands to

real :: arg1,arg2
double precision :: arg1, arg2
complex :: arg
double complex :: arg

I'd maybe also not name the dummy arguments in such a complicated way --- someone reading this code later on will be confused why there are so many underscores, and whether they are needed by f2py for something magical...

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Sorry forgot to look into this. Thanks for the pointer. This is still very murky to me. I still don't see why arg, \2 expands to arg2, arg, arg and why arg, arg should be assumed to be a complex number.

interface
function zselect(e_alpha__i__e,e_beta__i__e)
complex*16 :: e_alpha__i__e
complex*16 :: e_beta__i__e
logical :: zselect
end function zselect
end interface
end python module zgges__user__routines

python module dgges__user__routines
interface
function dselect(e_alphar__i__e,e_alphai__i__e,e_beta__i__e)
double precision :: e_alphar__i__e
double precision :: e_alphai__i__e
double precision :: e_beta__i__e
logical :: dselect
end function dselect
end interface
end python module dgges__user__routines

python module cgges__user__routines
interface
function cselect(e_alpha__i__e,e_beta__i__e)
complex :: e_alpha__i__e
complex :: e_beta__i__e
logical :: zselect
end function zselect
end interface
end python module cgges__user__routines

python module sgges__user__routines
interface
function sselect(e_alphar__i__e,e_alphai__i__e,e_beta__i__e)
real :: e_alphar__i__e
real :: e_alphai__i__e
real :: e_beta__i__e
logical :: dselect
end function dselect
end interface
end python module sgges__user__routines