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installed required libs
2021-02-17 12:26:31 +05:30
import math
import numpy as np
from numpy.lib.stride_tricks import as_strided
__all__ = ['tri', 'tril', 'triu', 'toeplitz', 'circulant', 'hankel',
'hadamard', 'leslie', 'kron', 'block_diag', 'companion',
'helmert', 'hilbert', 'invhilbert', 'pascal', 'invpascal', 'dft',
'fiedler', 'fiedler_companion', 'convolution_matrix']
# -----------------------------------------------------------------------------
# matrix construction functions
# -----------------------------------------------------------------------------
#
# *Note*: tri{,u,l} is implemented in NumPy, but an important bug was fixed in
# 2.0.0.dev-1af2f3, the following tri{,u,l} definitions are here for backwards
# compatibility.
def tri(N, M=None, k=0, dtype=None):
"""
Construct (N, M) matrix filled with ones at and below the kth diagonal.
The matrix has A[i,j] == 1 for i <= j + k
Parameters
----------
N : int
The size of the first dimension of the matrix.
M : int or None, optional
The size of the second dimension of the matrix. If `M` is None,
`M = N` is assumed.
k : int, optional
Number of subdiagonal below which matrix is filled with ones.
`k` = 0 is the main diagonal, `k` < 0 subdiagonal and `k` > 0
superdiagonal.
dtype : dtype, optional
Data type of the matrix.
Returns
-------
tri : (N, M) ndarray
Tri matrix.
Examples
--------
>>> from scipy.linalg import tri
>>> tri(3, 5, 2, dtype=int)
array([[1, 1, 1, 0, 0],
[1, 1, 1, 1, 0],
[1, 1, 1, 1, 1]])
>>> tri(3, 5, -1, dtype=int)
array([[0, 0, 0, 0, 0],
[1, 0, 0, 0, 0],
[1, 1, 0, 0, 0]])
"""
if M is None:
M = N
if isinstance(M, str):
# pearu: any objections to remove this feature?
# As tri(N,'d') is equivalent to tri(N,dtype='d')
dtype = M
M = N
m = np.greater_equal.outer(np.arange(k, N+k), np.arange(M))
if dtype is None:
return m
else:
return m.astype(dtype)
def tril(m, k=0):
"""
Make a copy of a matrix with elements above the kth diagonal zeroed.
Parameters
----------
m : array_like
Matrix whose elements to return
k : int, optional
Diagonal above which to zero elements.
`k` == 0 is the main diagonal, `k` < 0 subdiagonal and
`k` > 0 superdiagonal.
Returns
-------
tril : ndarray
Return is the same shape and type as `m`.
Examples
--------
>>> from scipy.linalg import tril
>>> tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 0, 0, 0],
[ 4, 0, 0],
[ 7, 8, 0],
[10, 11, 12]])
"""
m = np.asarray(m)
out = tri(m.shape[0], m.shape[1], k=k, dtype=m.dtype.char) * m
return out
def triu(m, k=0):
"""
Make a copy of a matrix with elements below the kth diagonal zeroed.
Parameters
----------
m : array_like
Matrix whose elements to return
k : int, optional
Diagonal below which to zero elements.
`k` == 0 is the main diagonal, `k` < 0 subdiagonal and
`k` > 0 superdiagonal.
Returns
-------
triu : ndarray
Return matrix with zeroed elements below the kth diagonal and has
same shape and type as `m`.
Examples
--------
>>> from scipy.linalg import triu
>>> triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
"""
m = np.asarray(m)
out = (1 - tri(m.shape[0], m.shape[1], k - 1, m.dtype.char)) * m
return out
def toeplitz(c, r=None):
"""
Construct a Toeplitz matrix.
The Toeplitz matrix has constant diagonals, with c as its first column
and r as its first row. If r is not given, ``r == conjugate(c)`` is
assumed.
Parameters
----------
c : array_like
First column of the matrix. Whatever the actual shape of `c`, it
will be converted to a 1-D array.
r : array_like, optional
First row of the matrix. If None, ``r = conjugate(c)`` is assumed;
in this case, if c[0] is real, the result is a Hermitian matrix.
r[0] is ignored; the first row of the returned matrix is
``[c[0], r[1:]]``. Whatever the actual shape of `r`, it will be
converted to a 1-D array.
Returns
-------
A : (len(c), len(r)) ndarray
The Toeplitz matrix. Dtype is the same as ``(c[0] + r[0]).dtype``.
See Also
--------
circulant : circulant matrix
hankel : Hankel matrix
solve_toeplitz : Solve a Toeplitz system.
Notes
-----
The behavior when `c` or `r` is a scalar, or when `c` is complex and
`r` is None, was changed in version 0.8.0. The behavior in previous
versions was undocumented and is no longer supported.
Examples
--------
>>> from scipy.linalg import toeplitz
>>> toeplitz([1,2,3], [1,4,5,6])
array([[1, 4, 5, 6],
[2, 1, 4, 5],
[3, 2, 1, 4]])
>>> toeplitz([1.0, 2+3j, 4-1j])
array([[ 1.+0.j, 2.-3.j, 4.+1.j],
[ 2.+3.j, 1.+0.j, 2.-3.j],
[ 4.-1.j, 2.+3.j, 1.+0.j]])
"""
c = np.asarray(c).ravel()
if r is None:
r = c.conjugate()
else:
r = np.asarray(r).ravel()
# Form a 1-D array containing a reversed c followed by r[1:] that could be
# strided to give us toeplitz matrix.
vals = np.concatenate((c[::-1], r[1:]))
out_shp = len(c), len(r)
n = vals.strides[0]
return as_strided(vals[len(c)-1:], shape=out_shp, strides=(-n, n)).copy()
def circulant(c):
"""
Construct a circulant matrix.
Parameters
----------
c : (N,) array_like
1-D array, the first column of the matrix.
Returns
-------
A : (N, N) ndarray
A circulant matrix whose first column is `c`.
See Also
--------
toeplitz : Toeplitz matrix
hankel : Hankel matrix
solve_circulant : Solve a circulant system.
Notes
-----
.. versionadded:: 0.8.0
Examples
--------
>>> from scipy.linalg import circulant
>>> circulant([1, 2, 3])
array([[1, 3, 2],
[2, 1, 3],
[3, 2, 1]])
"""
c = np.asarray(c).ravel()
# Form an extended array that could be strided to give circulant version
c_ext = np.concatenate((c[::-1], c[:0:-1]))
L = len(c)
n = c_ext.strides[0]
return as_strided(c_ext[L-1:], shape=(L, L), strides=(-n, n)).copy()
def hankel(c, r=None):
"""
Construct a Hankel matrix.
The Hankel matrix has constant anti-diagonals, with `c` as its
first column and `r` as its last row. If `r` is not given, then
`r = zeros_like(c)` is assumed.
Parameters
----------
c : array_like
First column of the matrix. Whatever the actual shape of `c`, it
will be converted to a 1-D array.
r : array_like, optional
Last row of the matrix. If None, ``r = zeros_like(c)`` is assumed.
r[0] is ignored; the last row of the returned matrix is
``[c[-1], r[1:]]``. Whatever the actual shape of `r`, it will be
converted to a 1-D array.
Returns
-------
A : (len(c), len(r)) ndarray
The Hankel matrix. Dtype is the same as ``(c[0] + r[0]).dtype``.
See Also
--------
toeplitz : Toeplitz matrix
circulant : circulant matrix
Examples
--------
>>> from scipy.linalg import hankel
>>> hankel([1, 17, 99])
array([[ 1, 17, 99],
[17, 99, 0],
[99, 0, 0]])
>>> hankel([1,2,3,4], [4,7,7,8,9])
array([[1, 2, 3, 4, 7],
[2, 3, 4, 7, 7],
[3, 4, 7, 7, 8],
[4, 7, 7, 8, 9]])
"""
c = np.asarray(c).ravel()
if r is None:
r = np.zeros_like(c)
else:
r = np.asarray(r).ravel()
# Form a 1-D array of values to be used in the matrix, containing `c`
# followed by r[1:].
vals = np.concatenate((c, r[1:]))
# Stride on concatenated array to get hankel matrix
out_shp = len(c), len(r)
n = vals.strides[0]
return as_strided(vals, shape=out_shp, strides=(n, n)).copy()
def hadamard(n, dtype=int):
"""
Construct an Hadamard matrix.
Constructs an n-by-n Hadamard matrix, using Sylvester's
construction. `n` must be a power of 2.
Parameters
----------
n : int
The order of the matrix. `n` must be a power of 2.
dtype : dtype, optional
The data type of the array to be constructed.
Returns
-------
H : (n, n) ndarray
The Hadamard matrix.
Notes
-----
.. versionadded:: 0.8.0
Examples
--------
>>> from scipy.linalg import hadamard
>>> hadamard(2, dtype=complex)
array([[ 1.+0.j, 1.+0.j],
[ 1.+0.j, -1.-0.j]])
>>> hadamard(4)
array([[ 1, 1, 1, 1],
[ 1, -1, 1, -1],
[ 1, 1, -1, -1],
[ 1, -1, -1, 1]])
"""
# This function is a slightly modified version of the
# function contributed by Ivo in ticket #675.
if n < 1:
lg2 = 0
else:
lg2 = int(math.log(n, 2))
if 2 ** lg2 != n:
raise ValueError("n must be an positive integer, and n must be "
"a power of 2")
H = np.array([[1]], dtype=dtype)
# Sylvester's construction
for i in range(0, lg2):
H = np.vstack((np.hstack((H, H)), np.hstack((H, -H))))
return H
def leslie(f, s):
"""
Create a Leslie matrix.
Given the length n array of fecundity coefficients `f` and the length
n-1 array of survival coefficients `s`, return the associated Leslie
matrix.
Parameters
----------
f : (N,) array_like
The "fecundity" coefficients.
s : (N-1,) array_like
The "survival" coefficients, has to be 1-D. The length of `s`
must be one less than the length of `f`, and it must be at least 1.
Returns
-------
L : (N, N) ndarray
The array is zero except for the first row,
which is `f`, and the first sub-diagonal, which is `s`.
The data-type of the array will be the data-type of ``f[0]+s[0]``.
Notes
-----
.. versionadded:: 0.8.0
The Leslie matrix is used to model discrete-time, age-structured
population growth [1]_ [2]_. In a population with `n` age classes, two sets
of parameters define a Leslie matrix: the `n` "fecundity coefficients",
which give the number of offspring per-capita produced by each age
class, and the `n` - 1 "survival coefficients", which give the
per-capita survival rate of each age class.
References
----------
.. [1] P. H. Leslie, On the use of matrices in certain population
mathematics, Biometrika, Vol. 33, No. 3, 183--212 (Nov. 1945)
.. [2] P. H. Leslie, Some further notes on the use of matrices in
population mathematics, Biometrika, Vol. 35, No. 3/4, 213--245
(Dec. 1948)
Examples
--------
>>> from scipy.linalg import leslie
>>> leslie([0.1, 2.0, 1.0, 0.1], [0.2, 0.8, 0.7])
array([[ 0.1, 2. , 1. , 0.1],
[ 0.2, 0. , 0. , 0. ],
[ 0. , 0.8, 0. , 0. ],
[ 0. , 0. , 0.7, 0. ]])
"""
f = np.atleast_1d(f)
s = np.atleast_1d(s)
if f.ndim != 1:
raise ValueError("Incorrect shape for f. f must be 1D")
if s.ndim != 1:
raise ValueError("Incorrect shape for s. s must be 1D")
if f.size != s.size + 1:
raise ValueError("Incorrect lengths for f and s. The length"
" of s must be one less than the length of f.")
if s.size == 0:
raise ValueError("The length of s must be at least 1.")
tmp = f[0] + s[0]
n = f.size
a = np.zeros((n, n), dtype=tmp.dtype)
a[0] = f
a[list(range(1, n)), list(range(0, n - 1))] = s
return a
def kron(a, b):
"""
Kronecker product.
The result is the block matrix::
a[0,0]*b a[0,1]*b ... a[0,-1]*b
a[1,0]*b a[1,1]*b ... a[1,-1]*b
...
a[-1,0]*b a[-1,1]*b ... a[-1,-1]*b
Parameters
----------
a : (M, N) ndarray
Input array
b : (P, Q) ndarray
Input array
Returns
-------
A : (M*P, N*Q) ndarray
Kronecker product of `a` and `b`.
Examples
--------
>>> from numpy import array
>>> from scipy.linalg import kron
>>> kron(array([[1,2],[3,4]]), array([[1,1,1]]))
array([[1, 1, 1, 2, 2, 2],
[3, 3, 3, 4, 4, 4]])
"""
if not a.flags['CONTIGUOUS']:
a = np.reshape(a, a.shape)
if not b.flags['CONTIGUOUS']:
b = np.reshape(b, b.shape)
o = np.outer(a, b)
o = o.reshape(a.shape + b.shape)
return np.concatenate(np.concatenate(o, axis=1), axis=1)
def block_diag(*arrs):
"""
Create a block diagonal matrix from provided arrays.
Given the inputs `A`, `B` and `C`, the output will have these
arrays arranged on the diagonal::
[[A, 0, 0],
[0, B, 0],
[0, 0, C]]
Parameters
----------
A, B, C, ... : array_like, up to 2-D
Input arrays. A 1-D array or array_like sequence of length `n` is
treated as a 2-D array with shape ``(1,n)``.
Returns
-------
D : ndarray
Array with `A`, `B`, `C`, ... on the diagonal. `D` has the
same dtype as `A`.
Notes
-----
If all the input arrays are square, the output is known as a
block diagonal matrix.
Empty sequences (i.e., array-likes of zero size) will not be ignored.
Noteworthy, both [] and [[]] are treated as matrices with shape ``(1,0)``.
Examples
--------
>>> from scipy.linalg import block_diag
>>> A = [[1, 0],
... [0, 1]]
>>> B = [[3, 4, 5],
... [6, 7, 8]]
>>> C = [[7]]
>>> P = np.zeros((2, 0), dtype='int32')
>>> block_diag(A, B, C)
array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 3, 4, 5, 0],
[0, 0, 6, 7, 8, 0],
[0, 0, 0, 0, 0, 7]])
>>> block_diag(A, P, B, C)
array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 3, 4, 5, 0],
[0, 0, 6, 7, 8, 0],
[0, 0, 0, 0, 0, 7]])
>>> block_diag(1.0, [2, 3], [[4, 5], [6, 7]])
array([[ 1., 0., 0., 0., 0.],
[ 0., 2., 3., 0., 0.],
[ 0., 0., 0., 4., 5.],
[ 0., 0., 0., 6., 7.]])
"""
if arrs == ():
arrs = ([],)
arrs = [np.atleast_2d(a) for a in arrs]
bad_args = [k for k in range(len(arrs)) if arrs[k].ndim > 2]
if bad_args:
raise ValueError("arguments in the following positions have dimension "
"greater than 2: %s" % bad_args)
shapes = np.array([a.shape for a in arrs])
out_dtype = np.find_common_type([arr.dtype for arr in arrs], [])
out = np.zeros(np.sum(shapes, axis=0), dtype=out_dtype)
r, c = 0, 0
for i, (rr, cc) in enumerate(shapes):
out[r:r + rr, c:c + cc] = arrs[i]
r += rr
c += cc
return out
def companion(a):
"""
Create a companion matrix.
Create the companion matrix [1]_ associated with the polynomial whose
coefficients are given in `a`.
Parameters
----------
a : (N,) array_like
1-D array of polynomial coefficients. The length of `a` must be
at least two, and ``a[0]`` must not be zero.
Returns
-------
c : (N-1, N-1) ndarray
The first row of `c` is ``-a[1:]/a[0]``, and the first
sub-diagonal is all ones. The data-type of the array is the same
as the data-type of ``1.0*a[0]``.
Raises
------
ValueError
If any of the following are true: a) ``a.ndim != 1``;
b) ``a.size < 2``; c) ``a[0] == 0``.
Notes
-----
.. versionadded:: 0.8.0
References
----------
.. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*. Cambridge, UK:
Cambridge University Press, 1999, pp. 146-7.
Examples
--------
>>> from scipy.linalg import companion
>>> companion([1, -10, 31, -30])
array([[ 10., -31., 30.],
[ 1., 0., 0.],
[ 0., 1., 0.]])
"""
a = np.atleast_1d(a)
if a.ndim != 1:
raise ValueError("Incorrect shape for `a`. `a` must be "
"one-dimensional.")
if a.size < 2:
raise ValueError("The length of `a` must be at least 2.")
if a[0] == 0:
raise ValueError("The first coefficient in `a` must not be zero.")
first_row = -a[1:] / (1.0 * a[0])
n = a.size
c = np.zeros((n - 1, n - 1), dtype=first_row.dtype)
c[0] = first_row
c[list(range(1, n - 1)), list(range(0, n - 2))] = 1
return c
def helmert(n, full=False):
"""
Create an Helmert matrix of order `n`.
This has applications in statistics, compositional or simplicial analysis,
and in Aitchison geometry.
Parameters
----------
n : int
The size of the array to create.
full : bool, optional
If True the (n, n) ndarray will be returned.
Otherwise the submatrix that does not include the first
row will be returned.
Default: False.
Returns
-------
M : ndarray
The Helmert matrix.
The shape is (n, n) or (n-1, n) depending on the `full` argument.
Examples
--------
>>> from scipy.linalg import helmert
>>> helmert(5, full=True)
array([[ 0.4472136 , 0.4472136 , 0.4472136 , 0.4472136 , 0.4472136 ],
[ 0.70710678, -0.70710678, 0. , 0. , 0. ],
[ 0.40824829, 0.40824829, -0.81649658, 0. , 0. ],
[ 0.28867513, 0.28867513, 0.28867513, -0.8660254 , 0. ],
[ 0.2236068 , 0.2236068 , 0.2236068 , 0.2236068 , -0.89442719]])
"""
H = np.tril(np.ones((n, n)), -1) - np.diag(np.arange(n))
d = np.arange(n) * np.arange(1, n+1)
H[0] = 1
d[0] = n
H_full = H / np.sqrt(d)[:, np.newaxis]
if full:
return H_full
else:
return H_full[1:]
def hilbert(n):
"""
Create a Hilbert matrix of order `n`.
Returns the `n` by `n` array with entries `h[i,j] = 1 / (i + j + 1)`.
Parameters
----------
n : int
The size of the array to create.
Returns
-------
h : (n, n) ndarray
The Hilbert matrix.
See Also
--------
invhilbert : Compute the inverse of a Hilbert matrix.
Notes
-----
.. versionadded:: 0.10.0
Examples
--------
>>> from scipy.linalg import hilbert
>>> hilbert(3)
array([[ 1. , 0.5 , 0.33333333],
[ 0.5 , 0.33333333, 0.25 ],
[ 0.33333333, 0.25 , 0.2 ]])
"""
values = 1.0 / (1.0 + np.arange(2 * n - 1))
h = hankel(values[:n], r=values[n - 1:])
return h
def invhilbert(n, exact=False):
"""
Compute the inverse of the Hilbert matrix of order `n`.
The entries in the inverse of a Hilbert matrix are integers. When `n`
is greater than 14, some entries in the inverse exceed the upper limit
of 64 bit integers. The `exact` argument provides two options for
dealing with these large integers.
Parameters
----------
n : int
The order of the Hilbert matrix.
exact : bool, optional
If False, the data type of the array that is returned is np.float64,
and the array is an approximation of the inverse.
If True, the array is the exact integer inverse array. To represent
the exact inverse when n > 14, the returned array is an object array
of long integers. For n <= 14, the exact inverse is returned as an
array with data type np.int64.
Returns
-------
invh : (n, n) ndarray
The data type of the array is np.float64 if `exact` is False.
If `exact` is True, the data type is either np.int64 (for n <= 14)
or object (for n > 14). In the latter case, the objects in the
array will be long integers.
See Also
--------
hilbert : Create a Hilbert matrix.
Notes
-----
.. versionadded:: 0.10.0
Examples
--------
>>> from scipy.linalg import invhilbert
>>> invhilbert(4)
array([[ 16., -120., 240., -140.],
[ -120., 1200., -2700., 1680.],
[ 240., -2700., 6480., -4200.],
[ -140., 1680., -4200., 2800.]])
>>> invhilbert(4, exact=True)
array([[ 16, -120, 240, -140],
[ -120, 1200, -2700, 1680],
[ 240, -2700, 6480, -4200],
[ -140, 1680, -4200, 2800]], dtype=int64)
>>> invhilbert(16)[7,7]
4.2475099528537506e+19
>>> invhilbert(16, exact=True)[7,7]
42475099528537378560
"""
from scipy.special import comb
if exact:
if n > 14:
dtype = object
else:
dtype = np.int64
else:
dtype = np.float64
invh = np.empty((n, n), dtype=dtype)
for i in range(n):
for j in range(0, i + 1):
s = i + j
invh[i, j] = ((-1) ** s * (s + 1) *
comb(n + i, n - j - 1, exact) *
comb(n + j, n - i - 1, exact) *
comb(s, i, exact) ** 2)
if i != j:
invh[j, i] = invh[i, j]
return invh
def pascal(n, kind='symmetric', exact=True):
"""
Returns the n x n Pascal matrix.
The Pascal matrix is a matrix containing the binomial coefficients as
its elements.
Parameters
----------
n : int
The size of the matrix to create; that is, the result is an n x n
matrix.
kind : str, optional
Must be one of 'symmetric', 'lower', or 'upper'.
Default is 'symmetric'.
exact : bool, optional
If `exact` is True, the result is either an array of type
numpy.uint64 (if n < 35) or an object array of Python long integers.
If `exact` is False, the coefficients in the matrix are computed using
`scipy.special.comb` with `exact=False`. The result will be a floating
point array, and the values in the array will not be the exact
coefficients, but this version is much faster than `exact=True`.
Returns
-------
p : (n, n) ndarray
The Pascal matrix.
See Also
--------
invpascal
Notes
-----
See https://en.wikipedia.org/wiki/Pascal_matrix for more information
about Pascal matrices.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy.linalg import pascal
>>> pascal(4)
array([[ 1, 1, 1, 1],
[ 1, 2, 3, 4],
[ 1, 3, 6, 10],
[ 1, 4, 10, 20]], dtype=uint64)
>>> pascal(4, kind='lower')
array([[1, 0, 0, 0],
[1, 1, 0, 0],
[1, 2, 1, 0],
[1, 3, 3, 1]], dtype=uint64)
>>> pascal(50)[-1, -1]
25477612258980856902730428600
>>> from scipy.special import comb
>>> comb(98, 49, exact=True)
25477612258980856902730428600
"""
from scipy.special import comb
if kind not in ['symmetric', 'lower', 'upper']:
raise ValueError("kind must be 'symmetric', 'lower', or 'upper'")
if exact:
if n >= 35:
L_n = np.empty((n, n), dtype=object)
L_n.fill(0)
else:
L_n = np.zeros((n, n), dtype=np.uint64)
for i in range(n):
for j in range(i + 1):
L_n[i, j] = comb(i, j, exact=True)
else:
L_n = comb(*np.ogrid[:n, :n])
if kind == 'lower':
p = L_n
elif kind == 'upper':
p = L_n.T
else:
p = np.dot(L_n, L_n.T)
return p
def invpascal(n, kind='symmetric', exact=True):
"""
Returns the inverse of the n x n Pascal matrix.
The Pascal matrix is a matrix containing the binomial coefficients as
its elements.
Parameters
----------
n : int
The size of the matrix to create; that is, the result is an n x n
matrix.
kind : str, optional
Must be one of 'symmetric', 'lower', or 'upper'.
Default is 'symmetric'.
exact : bool, optional
If `exact` is True, the result is either an array of type
``numpy.int64`` (if `n` <= 35) or an object array of Python integers.
If `exact` is False, the coefficients in the matrix are computed using
`scipy.special.comb` with `exact=False`. The result will be a floating
point array, and for large `n`, the values in the array will not be the
exact coefficients.
Returns
-------
invp : (n, n) ndarray
The inverse of the Pascal matrix.
See Also
--------
pascal
Notes
-----
.. versionadded:: 0.16.0
References
----------
.. [1] "Pascal matrix", https://en.wikipedia.org/wiki/Pascal_matrix
.. [2] Cohen, A. M., "The inverse of a Pascal matrix", Mathematical
Gazette, 59(408), pp. 111-112, 1975.
Examples
--------
>>> from scipy.linalg import invpascal, pascal
>>> invp = invpascal(5)
>>> invp
array([[ 5, -10, 10, -5, 1],
[-10, 30, -35, 19, -4],
[ 10, -35, 46, -27, 6],
[ -5, 19, -27, 17, -4],
[ 1, -4, 6, -4, 1]])
>>> p = pascal(5)
>>> p.dot(invp)
array([[ 1., 0., 0., 0., 0.],
[ 0., 1., 0., 0., 0.],
[ 0., 0., 1., 0., 0.],
[ 0., 0., 0., 1., 0.],
[ 0., 0., 0., 0., 1.]])
An example of the use of `kind` and `exact`:
>>> invpascal(5, kind='lower', exact=False)
array([[ 1., -0., 0., -0., 0.],
[-1., 1., -0., 0., -0.],
[ 1., -2., 1., -0., 0.],
[-1., 3., -3., 1., -0.],
[ 1., -4., 6., -4., 1.]])
"""
from scipy.special import comb
if kind not in ['symmetric', 'lower', 'upper']:
raise ValueError("'kind' must be 'symmetric', 'lower' or 'upper'.")
if kind == 'symmetric':
if exact:
if n > 34:
dt = object
else:
dt = np.int64
else:
dt = np.float64
invp = np.empty((n, n), dtype=dt)
for i in range(n):
for j in range(0, i + 1):
v = 0
for k in range(n - i):
v += comb(i + k, k, exact=exact) * comb(i + k, i + k - j,
exact=exact)
invp[i, j] = (-1)**(i - j) * v
if i != j:
invp[j, i] = invp[i, j]
else:
# For the 'lower' and 'upper' cases, we computer the inverse by
# changing the sign of every other diagonal of the pascal matrix.
invp = pascal(n, kind=kind, exact=exact)
if invp.dtype == np.uint64:
# This cast from np.uint64 to int64 OK, because if `kind` is not
# "symmetric", the values in invp are all much less than 2**63.
invp = invp.view(np.int64)
# The toeplitz matrix has alternating bands of 1 and -1.
invp *= toeplitz((-1)**np.arange(n)).astype(invp.dtype)
return invp
def dft(n, scale=None):
"""
Discrete Fourier transform matrix.
Create the matrix that computes the discrete Fourier transform of a
sequence [1]_. The nth primitive root of unity used to generate the
matrix is exp(-2*pi*i/n), where i = sqrt(-1).
Parameters
----------
n : int
Size the matrix to create.
scale : str, optional
Must be None, 'sqrtn', or 'n'.
If `scale` is 'sqrtn', the matrix is divided by `sqrt(n)`.
If `scale` is 'n', the matrix is divided by `n`.
If `scale` is None (the default), the matrix is not normalized, and the
return value is simply the Vandermonde matrix of the roots of unity.
Returns
-------
m : (n, n) ndarray
The DFT matrix.
Notes
-----
When `scale` is None, multiplying a vector by the matrix returned by
`dft` is mathematically equivalent to (but much less efficient than)
the calculation performed by `scipy.fft.fft`.
.. versionadded:: 0.14.0
References
----------
.. [1] "DFT matrix", https://en.wikipedia.org/wiki/DFT_matrix
Examples
--------
>>> from scipy.linalg import dft
>>> np.set_printoptions(precision=2, suppress=True) # for compact output
>>> m = dft(5)
>>> m
array([[ 1. +0.j , 1. +0.j , 1. +0.j , 1. +0.j , 1. +0.j ],
[ 1. +0.j , 0.31-0.95j, -0.81-0.59j, -0.81+0.59j, 0.31+0.95j],
[ 1. +0.j , -0.81-0.59j, 0.31+0.95j, 0.31-0.95j, -0.81+0.59j],
[ 1. +0.j , -0.81+0.59j, 0.31-0.95j, 0.31+0.95j, -0.81-0.59j],
[ 1. +0.j , 0.31+0.95j, -0.81+0.59j, -0.81-0.59j, 0.31-0.95j]])
>>> x = np.array([1, 2, 3, 0, 3])
>>> m @ x # Compute the DFT of x
array([ 9. +0.j , 0.12-0.81j, -2.12+3.44j, -2.12-3.44j, 0.12+0.81j])
Verify that ``m @ x`` is the same as ``fft(x)``.
>>> from scipy.fft import fft
>>> fft(x) # Same result as m @ x
array([ 9. +0.j , 0.12-0.81j, -2.12+3.44j, -2.12-3.44j, 0.12+0.81j])
"""
if scale not in [None, 'sqrtn', 'n']:
raise ValueError("scale must be None, 'sqrtn', or 'n'; "
"%r is not valid." % (scale,))
omegas = np.exp(-2j * np.pi * np.arange(n) / n).reshape(-1, 1)
m = omegas ** np.arange(n)
if scale == 'sqrtn':
m /= math.sqrt(n)
elif scale == 'n':
m /= n
return m
def fiedler(a):
"""Returns a symmetric Fiedler matrix
Given an sequence of numbers `a`, Fiedler matrices have the structure
``F[i, j] = np.abs(a[i] - a[j])``, and hence zero diagonals and nonnegative
entries. A Fiedler matrix has a dominant positive eigenvalue and other
eigenvalues are negative. Although not valid generally, for certain inputs,
the inverse and the determinant can be derived explicitly as given in [1]_.
Parameters
----------
a : (n,) array_like
coefficient array
Returns
-------
F : (n, n) ndarray
See Also
--------
circulant, toeplitz
Notes
-----
.. versionadded:: 1.3.0
References
----------
.. [1] J. Todd, "Basic Numerical Mathematics: Vol.2 : Numerical Algebra",
1977, Birkhauser, :doi:`10.1007/978-3-0348-7286-7`
Examples
--------
>>> from scipy.linalg import det, inv, fiedler
>>> a = [1, 4, 12, 45, 77]
>>> n = len(a)
>>> A = fiedler(a)
>>> A
array([[ 0, 3, 11, 44, 76],
[ 3, 0, 8, 41, 73],
[11, 8, 0, 33, 65],
[44, 41, 33, 0, 32],
[76, 73, 65, 32, 0]])
The explicit formulas for determinant and inverse seem to hold only for
monotonically increasing/decreasing arrays. Note the tridiagonal structure
and the corners.
>>> Ai = inv(A)
>>> Ai[np.abs(Ai) < 1e-12] = 0. # cleanup the numerical noise for display
>>> Ai
array([[-0.16008772, 0.16666667, 0. , 0. , 0.00657895],
[ 0.16666667, -0.22916667, 0.0625 , 0. , 0. ],
[ 0. , 0.0625 , -0.07765152, 0.01515152, 0. ],
[ 0. , 0. , 0.01515152, -0.03077652, 0.015625 ],
[ 0.00657895, 0. , 0. , 0.015625 , -0.00904605]])
>>> det(A)
15409151.999999998
>>> (-1)**(n-1) * 2**(n-2) * np.diff(a).prod() * (a[-1] - a[0])
15409152
"""
a = np.atleast_1d(a)
if a.ndim != 1:
raise ValueError("Input 'a' must be a 1D array.")
if a.size == 0:
return np.array([], dtype=float)
elif a.size == 1:
return np.array([[0.]])
else:
return np.abs(a[:, None] - a)
def fiedler_companion(a):
""" Returns a Fiedler companion matrix
Given a polynomial coefficient array ``a``, this function forms a
pentadiagonal matrix with a special structure whose eigenvalues coincides
with the roots of ``a``.
Parameters
----------
a : (N,) array_like
1-D array of polynomial coefficients in descending order with a nonzero
leading coefficient. For ``N < 2``, an empty array is returned.
Returns
-------
c : (N-1, N-1) ndarray
Resulting companion matrix
Notes
-----
Similar to `companion` the leading coefficient should be nonzero. In the case
the leading coefficient is not 1, other coefficients are rescaled before
the array generation. To avoid numerical issues, it is best to provide a
monic polynomial.
.. versionadded:: 1.3.0
See Also
--------
companion
References
----------
.. [1] M. Fiedler, " A note on companion matrices", Linear Algebra and its
Applications, 2003, :doi:`10.1016/S0024-3795(03)00548-2`
Examples
--------
>>> from scipy.linalg import fiedler_companion, eigvals
>>> p = np.poly(np.arange(1, 9, 2)) # [1., -16., 86., -176., 105.]
>>> fc = fiedler_companion(p)
>>> fc
array([[ 16., -86., 1., 0.],
[ 1., 0., 0., 0.],
[ 0., 176., 0., -105.],
[ 0., 1., 0., 0.]])
>>> eigvals(fc)
array([7.+0.j, 5.+0.j, 3.+0.j, 1.+0.j])
"""
a = np.atleast_1d(a)
if a.ndim != 1:
raise ValueError("Input 'a' must be a 1-D array.")
if a.size <= 2:
if a.size == 2:
return np.array([[-(a/a[0])[-1]]])
return np.array([], dtype=a.dtype)
if a[0] == 0.:
raise ValueError('Leading coefficient is zero.')
a = a/a[0]
n = a.size - 1
c = np.zeros((n, n), dtype=a.dtype)
# subdiagonals
c[range(3, n, 2), range(1, n-2, 2)] = 1.
c[range(2, n, 2), range(1, n-1, 2)] = -a[3::2]
# superdiagonals
c[range(0, n-2, 2), range(2, n, 2)] = 1.
c[range(0, n-1, 2), range(1, n, 2)] = -a[2::2]
c[[0, 1], 0] = [-a[1], 1]
return c
def convolution_matrix(a, n, mode='full'):
"""
Construct a convolution matrix.
Constructs the Toeplitz matrix representing one-dimensional
convolution [1]_. See the notes below for details.
Parameters
----------
a : (m,) array_like
The 1-D array to convolve.
n : int
The number of columns in the resulting matrix. It gives the length
of the input to be convolved with `a`. This is analogous to the
length of `v` in ``numpy.convolve(a, v)``.
mode : str
This is analogous to `mode` in ``numpy.convolve(v, a, mode)``.
It must be one of ('full', 'valid', 'same').
See below for how `mode` determines the shape of the result.
Returns
-------
A : (k, n) ndarray
The convolution matrix whose row count `k` depends on `mode`::
======= =========================
mode k
======= =========================
'full' m + n -1
'same' max(m, n)
'valid' max(m, n) - min(m, n) + 1
======= =========================
See Also
--------
toeplitz : Toeplitz matrix
Notes
-----
The code::
A = convolution_matrix(a, n, mode)
creates a Toeplitz matrix `A` such that ``A @ v`` is equivalent to
using ``convolve(a, v, mode)``. The returned array always has `n`
columns. The number of rows depends on the specified `mode`, as
explained above.
In the default 'full' mode, the entries of `A` are given by::
A[i, j] == (a[i-j] if (0 <= (i-j) < m) else 0)
where ``m = len(a)``. Suppose, for example, the input array is
``[x, y, z]``. The convolution matrix has the form::
[x, 0, 0, ..., 0, 0]
[y, x, 0, ..., 0, 0]
[z, y, x, ..., 0, 0]
...
[0, 0, 0, ..., x, 0]
[0, 0, 0, ..., y, x]
[0, 0, 0, ..., z, y]
[0, 0, 0, ..., 0, z]
In 'valid' mode, the entries of `A` are given by::
A[i, j] == (a[i-j+m-1] if (0 <= (i-j+m-1) < m) else 0)
This corresponds to a matrix whose rows are the subset of those from
the 'full' case where all the coefficients in `a` are contained in the
row. For input ``[x, y, z]``, this array looks like::
[z, y, x, 0, 0, ..., 0, 0, 0]
[0, z, y, x, 0, ..., 0, 0, 0]
[0, 0, z, y, x, ..., 0, 0, 0]
...
[0, 0, 0, 0, 0, ..., x, 0, 0]
[0, 0, 0, 0, 0, ..., y, x, 0]
[0, 0, 0, 0, 0, ..., z, y, x]
In the 'same' mode, the entries of `A` are given by::
d = (m - 1) // 2
A[i, j] == (a[i-j+d] if (0 <= (i-j+d) < m) else 0)
The typical application of the 'same' mode is when one has a signal of
length `n` (with `n` greater than ``len(a)``), and the desired output
is a filtered signal that is still of length `n`.
For input ``[x, y, z]``, this array looks like::
[y, x, 0, 0, ..., 0, 0, 0]
[z, y, x, 0, ..., 0, 0, 0]
[0, z, y, x, ..., 0, 0, 0]
[0, 0, z, y, ..., 0, 0, 0]
...
[0, 0, 0, 0, ..., y, x, 0]
[0, 0, 0, 0, ..., z, y, x]
[0, 0, 0, 0, ..., 0, z, y]
.. versionadded:: 1.5.0
References
----------
.. [1] "Convolution", https://en.wikipedia.org/wiki/Convolution
Examples
--------
>>> from scipy.linalg import convolution_matrix
>>> A = convolution_matrix([-1, 4, -2], 5, mode='same')
>>> A
array([[ 4, -1, 0, 0, 0],
[-2, 4, -1, 0, 0],
[ 0, -2, 4, -1, 0],
[ 0, 0, -2, 4, -1],
[ 0, 0, 0, -2, 4]])
Compare multiplication by `A` with the use of `numpy.convolve`.
>>> x = np.array([1, 2, 0, -3, 0.5])
>>> A @ x
array([ 2. , 6. , -1. , -12.5, 8. ])
Verify that ``A @ x`` produced the same result as applying the
convolution function.
>>> np.convolve([-1, 4, -2], x, mode='same')
array([ 2. , 6. , -1. , -12.5, 8. ])
For comparison to the case ``mode='same'`` shown above, here are the
matrices produced by ``mode='full'`` and ``mode='valid'`` for the
same coefficients and size.
>>> convolution_matrix([-1, 4, -2], 5, mode='full')
array([[-1, 0, 0, 0, 0],
[ 4, -1, 0, 0, 0],
[-2, 4, -1, 0, 0],
[ 0, -2, 4, -1, 0],
[ 0, 0, -2, 4, -1],
[ 0, 0, 0, -2, 4],
[ 0, 0, 0, 0, -2]])
>>> convolution_matrix([-1, 4, -2], 5, mode='valid')
array([[-2, 4, -1, 0, 0],
[ 0, -2, 4, -1, 0],
[ 0, 0, -2, 4, -1]])
"""
if n <= 0:
raise ValueError('n must be a positive integer.')
a = np.asarray(a)
if a.ndim != 1:
raise ValueError('convolution_matrix expects a one-dimensional '
'array as input')
if a.size == 0:
raise ValueError('len(a) must be at least 1.')
if mode not in ('full', 'valid', 'same'):
raise ValueError(
"'mode' argument must be one of ('full', 'valid', 'same')")
# create zero padded versions of the array
az = np.pad(a, (0, n-1), 'constant')
raz = np.pad(a[::-1], (0, n-1), 'constant')
if mode == 'same':
trim = min(n, len(a)) - 1
tb = trim//2
te = trim - tb
col0 = az[tb:len(az)-te]
row0 = raz[-n-tb:len(raz)-tb]
elif mode == 'valid':
tb = min(n, len(a)) - 1
te = tb
col0 = az[tb:len(az)-te]
row0 = raz[-n-tb:len(raz)-tb]
else: # 'full'
col0 = az
row0 = raz[-n:]
return toeplitz(col0, row0)