package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Create a view into the array with the given shape and strides.

.. warning:: This function has to be used with extreme care, see notes.

Parameters ---------- x : ndarray Array to create a new. shape : sequence of int, optional The shape of the new array. Defaults to ``x.shape``. strides : sequence of int, optional The strides of the new array. Defaults to ``x.strides``. subok : bool, optional .. versionadded:: 1.10

If True, subclasses are preserved. writeable : bool, optional .. versionadded:: 1.12

If set to False, the returned array will always be readonly. Otherwise it will be writable if the original array was. It is advisable to set this to False if possible (see Notes).

Returns ------- view : ndarray

See also -------- broadcast_to: broadcast an array to a given shape. reshape : reshape an array.

Notes ----- ``as_strided`` creates a view into the array given the exact strides and shape. This means it manipulates the internal data structure of ndarray and, if done incorrectly, the array elements can point to invalid memory and can corrupt results or crash your program. It is advisable to always use the original ``x.strides`` when calculating new strides to avoid reliance on a contiguous memory layout.

Furthermore, arrays created with this function often contain self overlapping memory, so that two elements are identical. Vectorized write operations on such arrays will typically be unpredictable. They may even give different results for small, large, or transposed arrays. Since writing to these arrays has to be tested and done with great care, you may want to use ``writeable=False`` to avoid accidental write operations.

For these reasons it is advisable to avoid ``as_strided`` when possible.

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.])

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])

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 ``a0`` must not be zero.

Returns ------- c : (N-1, N-1) ndarray The first row of `c` is ``-a1:/a0``, and the first sub-diagonal is all ones. The data-type of the array is the same as the data-type of ``1.0*a0``.

Raises ------ ValueError If any of the following are true: a) ``a.ndim != 1``; b) ``a.size < 2``; c) ``a0 == 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.])

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::

Ai, j == (ai-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::

Ai, j == (ai-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 Ai, j == (ai-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])

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)

Returns a symmetric Fiedler matrix

Given an sequence of numbers `a`, Fiedler matrices have the structure ``Fi, j = np.abs(ai - aj)``, 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) >>> Ainp.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 - a0) 15409152

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)

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])

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. r0 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 ``(c0 + r0).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])

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])

Create a Hilbert matrix of order `n`.

Returns the `n` by `n` array with entries `hi,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 ])

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

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.])

Kronecker product.

The result is the block matrix::

a0,0*b a0,1*b ... a0,-1*b a1,0*b a1,1*b ... a1,-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])

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 ``f0+s0``.

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. ])

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

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 c0 is real, the result is a Hermitian matrix. r0 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 ``(c0 + r0).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])

Construct (N, M) matrix filled with ones at and below the kth diagonal.

The matrix has Ai,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])

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])

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])