def diag_indices_from(arr): """ Return the indices to access the main diagonal of an n-dimensional array. See `diag_indices` for full details. Parameters ---------- arr : array, at least 2-D See Also -------- diag_indices Notes ----- .. versionadded:: 1.4.0 """ if not arr.ndim >= 2: raise ValueError("input array must be at least 2-d") # For more than d=2, the strided formula is only valid for arrays with # all dimensions equal, so we check first. if not alltrue(diff(arr.shape) == 0): raise ValueError("All dimensions of input must be of equal length") return diag_indices(arr.shape[0], arr.ndim)
def poly(seq_of_zeros): """ Return a sequence representing a polynomial given a sequence of roots. If the input is a matrix, return the characteristic polynomial. Example: >>> b = roots([1,3,1,5,6]) >>> poly(b) array([ 1., 3., 1., 5., 6.]) """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1]: seq_of_zeros = _eigvals(seq_of_zeros) elif len(sh) ==1: pass else: raise ValueError, "input must be 1d or square 2d array." if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate(sort_complex( NX.compress(roots.imag < 0,roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def poly(seq_of_zeros): """ Find the coefficients of a polynomial with the given sequence of roots. Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned. Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object. Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree: ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1. Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array). See Also -------- polyval : Evaluate a polynomial at a point. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.) The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by :math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`, where **I** is the `n`-by-`n` identity matrix. [2]_ References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996. .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980. Examples -------- Given a sequence of a polynomial's zeros: >>> np.poly((0, 0, 0)) # Multiple root example array([1, 0, 0, 0]) The line above represents z**3 + 0*z**2 + 0*z + 0. >>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ]) The line above represents z**3 - z/4 >>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) #random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([ 1. , 0. , 0.16666667]) Or a square matrix object: >>> np.poly(np.matrix(P)) array([ 1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: dt = seq_of_zeros.dtype # Let object arrays slip through, e.g. for arbitrary precision if dt != object: seq_of_zeros = seq_of_zeros.astype(mintypecode(dt.char)) else: raise ValueError("input must be 1d or non-empty square 2d array.") if len(seq_of_zeros) == 0: return 1.0 dt = seq_of_zeros.dtype a = ones((1, ), dtype=dt) for k in range(len(seq_of_zeros)): a = NX.convolve(a, array([1, -seq_of_zeros[k]], dtype=dt), mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate( sort_complex(NX.compress(roots.imag < 0, roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def poly(seq_of_zeros): """ Find the coefficients of a polynomial with the given sequence of roots. Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned. Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object. Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree: ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1. Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array). See Also -------- polyval : Evaluate a polynomial at a point. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.) The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by :math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`, where **I** is the `n`-by-`n` identity matrix. [2]_ References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996. .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980. Examples -------- Given a sequence of a polynomial's zeros: >>> np.poly((0, 0, 0)) # Multiple root example array([1, 0, 0, 0]) The line above represents z**3 + 0*z**2 + 0*z + 0. >>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ]) The line above represents z**3 - z/4 >>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) #random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([ 1. , 0. , 0.16666667]) Or a square matrix object: >>> np.poly(np.matrix(P)) array([ 1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: pass else: raise ValueError("input must be 1d or square 2d array.") if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate(sort_complex( NX.compress(roots.imag < 0,roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def __eq__(self, other): return NX.alltrue(self.coeffs == other.coeffs)
def fill_diagonal(a, val, wrap=False): """Fill the main diagonal of the given array of any dimensionality. For an array `a` with ``a.ndim >= 2``, the diagonal is the list of locations with indices ``a[i, ..., i]`` all identical. This function modifies the input array in-place, it does not return a value. Parameters ---------- a : array, at least 2-D. Array whose diagonal is to be filled, it gets modified in-place. val : scalar Value to be written on the diagonal, its type must be compatible with that of the array a. wrap : bool For tall matrices in NumPy version up to 1.6.2, the diagonal "wrapped" after N columns. You can have this behavior with this option. This affects only tall matrices. See also -------- diag_indices, diag_indices_from Notes ----- .. versionadded:: 1.4.0 This functionality can be obtained via `diag_indices`, but internally this version uses a much faster implementation that never constructs the indices and uses simple slicing. Examples -------- >>> a = np.zeros((3, 3), int) >>> np.fill_diagonal(a, 5) >>> a array([[5, 0, 0], [0, 5, 0], [0, 0, 5]]) The same function can operate on a 4-D array: >>> a = np.zeros((3, 3, 3, 3), int) >>> np.fill_diagonal(a, 4) We only show a few blocks for clarity: >>> a[0, 0] array([[4, 0, 0], [0, 0, 0], [0, 0, 0]]) >>> a[1, 1] array([[0, 0, 0], [0, 4, 0], [0, 0, 0]]) >>> a[2, 2] array([[0, 0, 0], [0, 0, 0], [0, 0, 4]]) The wrap option affects only tall matrices: >>> # tall matrices no wrap >>> a = np.zeros((5, 3),int) >>> fill_diagonal(a, 4) >>> a array([[4, 0, 0], [0, 4, 0], [0, 0, 4], [0, 0, 0], [0, 0, 0]]) >>> # tall matrices wrap >>> a = np.zeros((5, 3),int) >>> fill_diagonal(a, 4, wrap=True) >>> a array([[4, 0, 0], [0, 4, 0], [0, 0, 4], [0, 0, 0], [4, 0, 0]]) >>> # wide matrices >>> a = np.zeros((3, 5),int) >>> fill_diagonal(a, 4, wrap=True) >>> a array([[4, 0, 0, 0, 0], [0, 4, 0, 0, 0], [0, 0, 4, 0, 0]]) """ if a.ndim < 2: raise ValueError("array must be at least 2-d") end = None if a.ndim == 2: # Explicit, fast formula for the common case. For 2-d arrays, we # accept rectangular ones. step = a.shape[1] + 1 #This is needed to don't have tall matrix have the diagonal wrap. if not wrap: end = a.shape[1] * a.shape[1] else: # For more than d=2, the strided formula is only valid for arrays with # all dimensions equal, so we check first. if not alltrue(diff(a.shape) == 0): raise ValueError("All dimensions of input must be of equal length") step = 1 + (cumprod(a.shape[:-1])).sum() # Write the value out into the diagonal. a.flat[:end:step] = val
def fill_diagonal(a, val, wrap=False): """Fill the main diagonal of the given array of any dimensionality. For an array `a` with ``a.ndim > 2``, the diagonal is the list of locations with indices ``a[i, i, ..., i]`` all identical. This function modifies the input array in-place, it does not return a value. Parameters ---------- a : array, at least 2-D. Array whose diagonal is to be filled, it gets modified in-place. val : scalar Value to be written on the diagonal, its type must be compatible with that of the array a. wrap: bool For tall matrices in NumPy version up to 1.6.2, the diagonal "wrapped" after N columns. You can have this behavior with this option. This affect only tall matrices. See also -------- diag_indices, diag_indices_from Notes ----- .. versionadded:: 1.4.0 This functionality can be obtained via `diag_indices`, but internally this version uses a much faster implementation that never constructs the indices and uses simple slicing. Examples -------- >>> a = np.zeros((3, 3), int) >>> np.fill_diagonal(a, 5) >>> a array([[5, 0, 0], [0, 5, 0], [0, 0, 5]]) The same function can operate on a 4-D array: >>> a = np.zeros((3, 3, 3, 3), int) >>> np.fill_diagonal(a, 4) We only show a few blocks for clarity: >>> a[0, 0] array([[4, 0, 0], [0, 0, 0], [0, 0, 0]]) >>> a[1, 1] array([[0, 0, 0], [0, 4, 0], [0, 0, 0]]) >>> a[2, 2] array([[0, 0, 0], [0, 0, 0], [0, 0, 4]]) # tall matrices no wrap >>> a = np.zeros((5, 3),int) >>> fill_diagonal(a, 4) array([[4, 0, 0], [0, 4, 0], [0, 0, 4], [0, 0, 0], [0, 0, 0]]) # tall matrices wrap >>> a = np.zeros((5, 3),int) >>> fill_diagonal(a, 4) array([[4, 0, 0], [0, 4, 0], [0, 0, 4], [0, 0, 0], [4, 0, 0]]) # wide matrices >>> a = np.zeros((3, 5),int) >>> fill_diagonal(a, 4) array([[4, 0, 0, 0, 0], [0, 4, 0, 0, 0], [0, 0, 4, 0, 0]]) """ if a.ndim < 2: raise ValueError("array must be at least 2-d") end = None if a.ndim == 2: # Explicit, fast formula for the common case. For 2-d arrays, we # accept rectangular ones. step = a.shape[1] + 1 # This is needed to don't have tall matrix have the diagonal wrap. if not wrap: end = a.shape[1] * a.shape[1] else: # For more than d=2, the strided formula is only valid for arrays with # all dimensions equal, so we check first. if not alltrue(diff(a.shape) == 0): raise ValueError("All dimensions of input must be of equal length") step = 1 + (cumprod(a.shape[:-1])).sum() # Write the value out into the diagonal. a.flat[:end:step] = val
def poly(seq_of_zeros): """ Return polynomial coefficients given a sequence of roots. Calculate the coefficients of a polynomial given the zeros of the polynomial. If a square matrix is given, then the coefficients for characteristic equation of the matrix, defined by :math:`\\mathrm{det}(\\mathbf{A} - \\lambda \\mathbf{I})`, are returned. Parameters ---------- seq_of_zeros : ndarray A sequence of polynomial roots or a square matrix. Returns ------- coefs : ndarray A sequence of polynomial coefficients representing the polynomial :math:`\\mathrm{coefs}[0] x^{n-1} + \\mathrm{coefs}[1] x^{n-2} + ... + \\mathrm{coefs}[2] x + \\mathrm{coefs}[n]` See Also -------- numpy.poly1d : A one-dimensional polynomial class. numpy.roots : Return the roots of the polynomial coefficients in p numpy.polyfit : Least squares polynomial fit Examples -------- Given a sequence of polynomial zeros, >>> b = np.roots([1, 3, 1, 5, 6]) >>> np.poly(b) array([ 1., 3., 1., 5., 6.]) Given a square matrix, >>> P = np.array([[19, 3], [-2, 26]]) >>> np.poly(P) array([ 1., -45., 500.]) """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1]: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) ==1: pass else: raise ValueError, "input must be 1d or square 2d array." if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate(sort_complex( NX.compress(roots.imag < 0,roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def poly(seq_of_zeros): """ Return polynomial coefficients given a sequence of roots. Calculate the coefficients of a polynomial given the zeros of the polynomial. If a square matrix is given, then the coefficients for characteristic equation of the matrix, defined by :math:`\\mathrm{det}(\\mathbf{A} - \\lambda \\mathbf{I})`, are returned. Parameters ---------- seq_of_zeros : ndarray A sequence of polynomial roots or a square matrix. Returns ------- coefs : ndarray A sequence of polynomial coefficients representing the polynomial :math:`\\mathrm{coefs}[0] x^{n-1} + \\mathrm{coefs}[1] x^{n-2} + ... + \\mathrm{coefs}[2] x + \\mathrm{coefs}[n]` See Also -------- numpy.poly1d : A one-dimensional polynomial class. numpy.roots : Return the roots of the polynomial coefficients in p numpy.polyfit : Least squares polynomial fit Examples -------- Given a sequence of polynomial zeros, >>> b = np.roots([1, 3, 1, 5, 6]) >>> np.poly(b) array([ 1., 3., 1., 5., 6.]) Given a square matrix, >>> P = np.array([[19, 3], [-2, 26]]) >>> np.poly(P) array([ 1., -45., 500.]) """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1]: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: pass else: raise ValueError, "input must be 1d or square 2d array." if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate( sort_complex(NX.compress(roots.imag < 0, roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def fill_diagonal(a, val): """Fill the main diagonal of the given array of any dimensionality. For an array with ndim > 2, the diagonal is the list of locations with indices a[i,i,...,i], all identical. This function modifies the input array in-place, it does not return a value. This functionality can be obtained via diag_indices(), but internally this version uses a much faster implementation that never constructs the indices and uses simple slicing. Parameters ---------- a : array, at least 2-dimensional. Array whose diagonal is to be filled, it gets modified in-place. val : scalar Value to be written on the diagonal, its type must be compatible with that of the array a. See also -------- diag_indices, diag_indices_from Notes ----- .. versionadded:: 1.4.0 Examples -------- >>> a = zeros((3,3),int) >>> fill_diagonal(a,5) >>> a array([[5, 0, 0], [0, 5, 0], [0, 0, 5]]) The same function can operate on a 4-d array: >>> a = zeros((3,3,3,3),int) >>> fill_diagonal(a,4) We only show a few blocks for clarity: >>> a[0,0] array([[4, 0, 0], [0, 0, 0], [0, 0, 0]]) >>> a[1,1] array([[0, 0, 0], [0, 4, 0], [0, 0, 0]]) >>> a[2,2] array([[0, 0, 0], [0, 0, 0], [0, 0, 4]]) """ if a.ndim < 2: raise ValueError("array must be at least 2-d") if a.ndim == 2: # Explicit, fast formula for the common case. For 2-d arrays, we # accept rectangular ones. step = a.shape[1] + 1 else: # For more than d=2, the strided formula is only valid for arrays with # all dimensions equal, so we check first. if not alltrue(diff(a.shape) == 0): raise ValueError("All dimensions of input must be of equal length") step = 1 + (cumprod(a.shape[:-1])).sum() # Write the value out into the diagonal. a.flat[::step] = val