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| #!/usr/bin/python | |
| # -*- coding: utf-8 -*- | |
| ################################################################################################## | |
| # module for the symmetric eigenvalue problem | |
| # Copyright 2013 Timo Hartmann (thartmann15 at gmail.com) | |
| # | |
| # todo: | |
| # - implement balancing | |
| # | |
| ################################################################################################## | |
| """ | |
| The symmetric eigenvalue problem. | |
| --------------------------------- | |
| This file contains routines for the symmetric eigenvalue problem. | |
| high level routines: | |
| eigsy : real symmetric (ordinary) eigenvalue problem | |
| eighe : complex hermitian (ordinary) eigenvalue problem | |
| eigh : unified interface for eigsy and eighe | |
| svd_r : singular value decomposition for real matrices | |
| svd_c : singular value decomposition for complex matrices | |
| svd : unified interface for svd_r and svd_c | |
| low level routines: | |
| r_sy_tridiag : reduction of real symmetric matrix to real symmetric tridiagonal matrix | |
| c_he_tridiag_0 : reduction of complex hermitian matrix to real symmetric tridiagonal matrix | |
| c_he_tridiag_1 : auxiliary routine to c_he_tridiag_0 | |
| c_he_tridiag_2 : auxiliary routine to c_he_tridiag_0 | |
| tridiag_eigen : solves the real symmetric tridiagonal matrix eigenvalue problem | |
| svd_r_raw : raw singular value decomposition for real matrices | |
| svd_c_raw : raw singular value decomposition for complex matrices | |
| """ | |
| from ..libmp.backend import xrange | |
| from .eigen import defun | |
| def r_sy_tridiag(ctx, A, D, E, calc_ev = True): | |
| """ | |
| This routine transforms a real symmetric matrix A to a real symmetric | |
| tridiagonal matrix T using an orthogonal similarity transformation: | |
| Q' * A * Q = T (here ' denotes the matrix transpose). | |
| The orthogonal matrix Q is build up from Householder reflectors. | |
| parameters: | |
| A (input/output) On input, A contains the real symmetric matrix of | |
| dimension (n,n). On output, if calc_ev is true, A contains the | |
| orthogonal matrix Q, otherwise A is destroyed. | |
| D (output) real array of length n, contains the diagonal elements | |
| of the tridiagonal matrix | |
| E (output) real array of length n, contains the offdiagonal elements | |
| of the tridiagonal matrix in E[0:(n-1)] where is the dimension of | |
| the matrix A. E[n-1] is undefined. | |
| calc_ev (input) If calc_ev is true, this routine explicitly calculates the | |
| orthogonal matrix Q which is then returned in A. If calc_ev is | |
| false, Q is not explicitly calculated resulting in a shorter run time. | |
| This routine is a python translation of the fortran routine tred2.f in the | |
| software library EISPACK (see netlib.org) which itself is based on the algol | |
| procedure tred2 described in: | |
| - Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson | |
| - Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971) | |
| For a good introduction to Householder reflections, see also | |
| Stoer, Bulirsch - Introduction to Numerical Analysis. | |
| """ | |
| # note : the vector v of the i-th houshoulder reflector is stored in a[(i+1):,i] | |
| # whereas v/<v,v> is stored in a[i,(i+1):] | |
| n = A.rows | |
| for i in xrange(n - 1, 0, -1): | |
| # scale the vector | |
| scale = 0 | |
| for k in xrange(0, i): | |
| scale += abs(A[k,i]) | |
| scale_inv = 0 | |
| if scale != 0: | |
| scale_inv = 1/scale | |
| # sadly there are floating point numbers not equal to zero whose reciprocal is infinity | |
| if i == 1 or scale == 0 or ctx.isinf(scale_inv): | |
| E[i] = A[i-1,i] # nothing to do | |
| D[i] = 0 | |
| continue | |
| # calculate parameters for housholder transformation | |
| H = 0 | |
| for k in xrange(0, i): | |
| A[k,i] *= scale_inv | |
| H += A[k,i] * A[k,i] | |
| F = A[i-1,i] | |
| G = ctx.sqrt(H) | |
| if F > 0: | |
| G = -G | |
| E[i] = scale * G | |
| H -= F * G | |
| A[i-1,i] = F - G | |
| F = 0 | |
| # apply housholder transformation | |
| for j in xrange(0, i): | |
| if calc_ev: | |
| A[i,j] = A[j,i] / H | |
| G = 0 # calculate A*U | |
| for k in xrange(0, j + 1): | |
| G += A[k,j] * A[k,i] | |
| for k in xrange(j + 1, i): | |
| G += A[j,k] * A[k,i] | |
| E[j] = G / H # calculate P | |
| F += E[j] * A[j,i] | |
| HH = F / (2 * H) | |
| for j in xrange(0, i): # calculate reduced A | |
| F = A[j,i] | |
| G = E[j] - HH * F # calculate Q | |
| E[j] = G | |
| for k in xrange(0, j + 1): | |
| A[k,j] -= F * E[k] + G * A[k,i] | |
| D[i] = H | |
| for i in xrange(1, n): # better for compatibility | |
| E[i-1] = E[i] | |
| E[n-1] = 0 | |
| if calc_ev: | |
| D[0] = 0 | |
| for i in xrange(0, n): | |
| if D[i] != 0: | |
| for j in xrange(0, i): # accumulate transformation matrices | |
| G = 0 | |
| for k in xrange(0, i): | |
| G += A[i,k] * A[k,j] | |
| for k in xrange(0, i): | |
| A[k,j] -= G * A[k,i] | |
| D[i] = A[i,i] | |
| A[i,i] = 1 | |
| for j in xrange(0, i): | |
| A[j,i] = A[i,j] = 0 | |
| else: | |
| for i in xrange(0, n): | |
| D[i] = A[i,i] | |
| def c_he_tridiag_0(ctx, A, D, E, T): | |
| """ | |
| This routine transforms a complex hermitian matrix A to a real symmetric | |
| tridiagonal matrix T using an unitary similarity transformation: | |
| Q' * A * Q = T (here ' denotes the hermitian matrix transpose, | |
| i.e. transposition und conjugation). | |
| The unitary matrix Q is build up from Householder reflectors and | |
| an unitary diagonal matrix. | |
| parameters: | |
| A (input/output) On input, A contains the complex hermitian matrix | |
| of dimension (n,n). On output, A contains the unitary matrix Q | |
| in compressed form. | |
| D (output) real array of length n, contains the diagonal elements | |
| of the tridiagonal matrix. | |
| E (output) real array of length n, contains the offdiagonal elements | |
| of the tridiagonal matrix in E[0:(n-1)] where is the dimension of | |
| the matrix A. E[n-1] is undefined. | |
| T (output) complex array of length n, contains a unitary diagonal | |
| matrix. | |
| This routine is a python translation (in slightly modified form) of the fortran | |
| routine htridi.f in the software library EISPACK (see netlib.org) which itself | |
| is a complex version of the algol procedure tred1 described in: | |
| - Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson | |
| - Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971) | |
| For a good introduction to Householder reflections, see also | |
| Stoer, Bulirsch - Introduction to Numerical Analysis. | |
| """ | |
| n = A.rows | |
| T[n-1] = 1 | |
| for i in xrange(n - 1, 0, -1): | |
| # scale the vector | |
| scale = 0 | |
| for k in xrange(0, i): | |
| scale += abs(ctx.re(A[k,i])) + abs(ctx.im(A[k,i])) | |
| scale_inv = 0 | |
| if scale != 0: | |
| scale_inv = 1 / scale | |
| # sadly there are floating point numbers not equal to zero whose reciprocal is infinity | |
| if scale == 0 or ctx.isinf(scale_inv): | |
| E[i] = 0 | |
| D[i] = 0 | |
| T[i-1] = 1 | |
| continue | |
| if i == 1: | |
| F = A[i-1,i] | |
| f = abs(F) | |
| E[i] = f | |
| D[i] = 0 | |
| if f != 0: | |
| T[i-1] = T[i] * F / f | |
| else: | |
| T[i-1] = T[i] | |
| continue | |
| # calculate parameters for housholder transformation | |
| H = 0 | |
| for k in xrange(0, i): | |
| A[k,i] *= scale_inv | |
| rr = ctx.re(A[k,i]) | |
| ii = ctx.im(A[k,i]) | |
| H += rr * rr + ii * ii | |
| F = A[i-1,i] | |
| f = abs(F) | |
| G = ctx.sqrt(H) | |
| H += G * f | |
| E[i] = scale * G | |
| if f != 0: | |
| F = F / f | |
| TZ = - T[i] * F # T[i-1]=-T[i]*F, but we need T[i-1] as temporary storage | |
| G *= F | |
| else: | |
| TZ = -T[i] # T[i-1]=-T[i] | |
| A[i-1,i] += G | |
| F = 0 | |
| # apply housholder transformation | |
| for j in xrange(0, i): | |
| A[i,j] = A[j,i] / H | |
| G = 0 # calculate A*U | |
| for k in xrange(0, j + 1): | |
| G += ctx.conj(A[k,j]) * A[k,i] | |
| for k in xrange(j + 1, i): | |
| G += A[j,k] * A[k,i] | |
| T[j] = G / H # calculate P | |
| F += ctx.conj(T[j]) * A[j,i] | |
| HH = F / (2 * H) | |
| for j in xrange(0, i): # calculate reduced A | |
| F = A[j,i] | |
| G = T[j] - HH * F # calculate Q | |
| T[j] = G | |
| for k in xrange(0, j + 1): | |
| A[k,j] -= ctx.conj(F) * T[k] + ctx.conj(G) * A[k,i] | |
| # as we use the lower left part for storage | |
| # we have to use the transpose of the normal formula | |
| T[i-1] = TZ | |
| D[i] = H | |
| for i in xrange(1, n): # better for compatibility | |
| E[i-1] = E[i] | |
| E[n-1] = 0 | |
| D[0] = 0 | |
| for i in xrange(0, n): | |
| zw = D[i] | |
| D[i] = ctx.re(A[i,i]) | |
| A[i,i] = zw | |
| def c_he_tridiag_1(ctx, A, T): | |
| """ | |
| This routine forms the unitary matrix Q described in c_he_tridiag_0. | |
| parameters: | |
| A (input/output) On input, A is the same matrix as delivered by | |
| c_he_tridiag_0. On output, A is set to Q. | |
| T (input) On input, T is the same array as delivered by c_he_tridiag_0. | |
| """ | |
| n = A.rows | |
| for i in xrange(0, n): | |
| if A[i,i] != 0: | |
| for j in xrange(0, i): | |
| G = 0 | |
| for k in xrange(0, i): | |
| G += ctx.conj(A[i,k]) * A[k,j] | |
| for k in xrange(0, i): | |
| A[k,j] -= G * A[k,i] | |
| A[i,i] = 1 | |
| for j in xrange(0, i): | |
| A[j,i] = A[i,j] = 0 | |
| for i in xrange(0, n): | |
| for k in xrange(0, n): | |
| A[i,k] *= T[k] | |
| def c_he_tridiag_2(ctx, A, T, B): | |
| """ | |
| This routine applied the unitary matrix Q described in c_he_tridiag_0 | |
| onto the the matrix B, i.e. it forms Q*B. | |
| parameters: | |
| A (input) On input, A is the same matrix as delivered by c_he_tridiag_0. | |
| T (input) On input, T is the same array as delivered by c_he_tridiag_0. | |
| B (input/output) On input, B is a complex matrix. On output B is replaced | |
| by Q*B. | |
| This routine is a python translation of the fortran routine htribk.f in the | |
| software library EISPACK (see netlib.org). See c_he_tridiag_0 for more | |
| references. | |
| """ | |
| n = A.rows | |
| for i in xrange(0, n): | |
| for k in xrange(0, n): | |
| B[k,i] *= T[k] | |
| for i in xrange(0, n): | |
| if A[i,i] != 0: | |
| for j in xrange(0, n): | |
| G = 0 | |
| for k in xrange(0, i): | |
| G += ctx.conj(A[i,k]) * B[k,j] | |
| for k in xrange(0, i): | |
| B[k,j] -= G * A[k,i] | |
| def tridiag_eigen(ctx, d, e, z = False): | |
| """ | |
| This subroutine find the eigenvalues and the first components of the | |
| eigenvectors of a real symmetric tridiagonal matrix using the implicit | |
| QL method. | |
| parameters: | |
| d (input/output) real array of length n. on input, d contains the diagonal | |
| elements of the input matrix. on output, d contains the eigenvalues in | |
| ascending order. | |
| e (input) real array of length n. on input, e contains the offdiagonal | |
| elements of the input matrix in e[0:(n-1)]. On output, e has been | |
| destroyed. | |
| z (input/output) If z is equal to False, no eigenvectors will be computed. | |
| Otherwise on input z should have the format z[0:m,0:n] (i.e. a real or | |
| complex matrix of dimension (m,n) ). On output this matrix will be | |
| multiplied by the matrix of the eigenvectors (i.e. the columns of this | |
| matrix are the eigenvectors): z --> z*EV | |
| That means if z[i,j]={1 if j==j; 0 otherwise} on input, then on output | |
| z will contain the first m components of the eigenvectors. That means | |
| if m is equal to n, the i-th eigenvector will be z[:,i]. | |
| This routine is a python translation (in slightly modified form) of the | |
| fortran routine imtql2.f in the software library EISPACK (see netlib.org) | |
| which itself is based on the algol procudure imtql2 desribed in: | |
| - num. math. 12, p. 377-383(1968) by matrin and wilkinson | |
| - modified in num. math. 15, p. 450(1970) by dubrulle | |
| - handbook for auto. comp., vol. II-linear algebra, p. 241-248 (1971) | |
| See also the routine gaussq.f in netlog.org or acm algorithm 726. | |
| """ | |
| n = len(d) | |
| e[n-1] = 0 | |
| iterlim = 2 * ctx.dps | |
| for l in xrange(n): | |
| j = 0 | |
| while 1: | |
| m = l | |
| while 1: | |
| # look for a small subdiagonal element | |
| if m + 1 == n: | |
| break | |
| if abs(e[m]) <= ctx.eps * (abs(d[m]) + abs(d[m + 1])): | |
| break | |
| m = m + 1 | |
| if m == l: | |
| break | |
| if j >= iterlim: | |
| raise RuntimeError("tridiag_eigen: no convergence to an eigenvalue after %d iterations" % iterlim) | |
| j += 1 | |
| # form shift | |
| p = d[l] | |
| g = (d[l + 1] - p) / (2 * e[l]) | |
| r = ctx.hypot(g, 1) | |
| if g < 0: | |
| s = g - r | |
| else: | |
| s = g + r | |
| g = d[m] - p + e[l] / s | |
| s, c, p = 1, 1, 0 | |
| for i in xrange(m - 1, l - 1, -1): | |
| f = s * e[i] | |
| b = c * e[i] | |
| if abs(f) > abs(g): # this here is a slight improvement also used in gaussq.f or acm algorithm 726. | |
| c = g / f | |
| r = ctx.hypot(c, 1) | |
| e[i + 1] = f * r | |
| s = 1 / r | |
| c = c * s | |
| else: | |
| s = f / g | |
| r = ctx.hypot(s, 1) | |
| e[i + 1] = g * r | |
| c = 1 / r | |
| s = s * c | |
| g = d[i + 1] - p | |
| r = (d[i] - g) * s + 2 * c * b | |
| p = s * r | |
| d[i + 1] = g + p | |
| g = c * r - b | |
| if not isinstance(z, bool): | |
| # calculate eigenvectors | |
| for w in xrange(z.rows): | |
| f = z[w,i+1] | |
| z[w,i+1] = s * z[w,i] + c * f | |
| z[w,i ] = c * z[w,i] - s * f | |
| d[l] = d[l] - p | |
| e[l] = g | |
| e[m] = 0 | |
| for ii in xrange(1, n): | |
| # sort eigenvalues and eigenvectors (bubble-sort) | |
| i = ii - 1 | |
| k = i | |
| p = d[i] | |
| for j in xrange(ii, n): | |
| if d[j] >= p: | |
| continue | |
| k = j | |
| p = d[k] | |
| if k == i: | |
| continue | |
| d[k] = d[i] | |
| d[i] = p | |
| if not isinstance(z, bool): | |
| for w in xrange(z.rows): | |
| p = z[w,i] | |
| z[w,i] = z[w,k] | |
| z[w,k] = p | |
| ######################################################################################## | |
| def eigsy(ctx, A, eigvals_only = False, overwrite_a = False): | |
| """ | |
| This routine solves the (ordinary) eigenvalue problem for a real symmetric | |
| square matrix A. Given A, an orthogonal matrix Q is calculated which | |
| diagonalizes A: | |
| Q' A Q = diag(E) and Q Q' = Q' Q = 1 | |
| Here diag(E) is a diagonal matrix whose diagonal is E. | |
| ' denotes the transpose. | |
| The columns of Q are the eigenvectors of A and E contains the eigenvalues: | |
| A Q[:,i] = E[i] Q[:,i] | |
| input: | |
| A: real matrix of format (n,n) which is symmetric | |
| (i.e. A=A' or A[i,j]=A[j,i]) | |
| eigvals_only: if true, calculates only the eigenvalues E. | |
| if false, calculates both eigenvectors and eigenvalues. | |
| overwrite_a: if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| E: vector of format (n). contains the eigenvalues of A in ascending order. | |
| Q: orthogonal matrix of format (n,n). contains the eigenvectors | |
| of A as columns. | |
| return value: | |
| E if eigvals_only is true | |
| (E, Q) if eigvals_only is false | |
| example: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[3, 2], [2, 0]]) | |
| >>> E = mp.eigsy(A, eigvals_only = True) | |
| >>> print(E) | |
| [-1.0] | |
| [ 4.0] | |
| >>> A = mp.matrix([[1, 2], [2, 3]]) | |
| >>> E, Q = mp.eigsy(A) | |
| >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) | |
| [0.0] | |
| [0.0] | |
| see also: eighe, eigh, eig | |
| """ | |
| if not overwrite_a: | |
| A = A.copy() | |
| d = ctx.zeros(A.rows, 1) | |
| e = ctx.zeros(A.rows, 1) | |
| if eigvals_only: | |
| r_sy_tridiag(ctx, A, d, e, calc_ev = False) | |
| tridiag_eigen(ctx, d, e, False) | |
| return d | |
| else: | |
| r_sy_tridiag(ctx, A, d, e, calc_ev = True) | |
| tridiag_eigen(ctx, d, e, A) | |
| return (d, A) | |
| def eighe(ctx, A, eigvals_only = False, overwrite_a = False): | |
| """ | |
| This routine solves the (ordinary) eigenvalue problem for a complex | |
| hermitian square matrix A. Given A, an unitary matrix Q is calculated which | |
| diagonalizes A: | |
| Q' A Q = diag(E) and Q Q' = Q' Q = 1 | |
| Here diag(E) a is diagonal matrix whose diagonal is E. | |
| ' denotes the hermitian transpose (i.e. ordinary transposition and | |
| complex conjugation). | |
| The columns of Q are the eigenvectors of A and E contains the eigenvalues: | |
| A Q[:,i] = E[i] Q[:,i] | |
| input: | |
| A: complex matrix of format (n,n) which is hermitian | |
| (i.e. A=A' or A[i,j]=conj(A[j,i])) | |
| eigvals_only: if true, calculates only the eigenvalues E. | |
| if false, calculates both eigenvectors and eigenvalues. | |
| overwrite_a: if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| E: vector of format (n). contains the eigenvalues of A in ascending order. | |
| Q: unitary matrix of format (n,n). contains the eigenvectors | |
| of A as columns. | |
| return value: | |
| E if eigvals_only is true | |
| (E, Q) if eigvals_only is false | |
| example: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[1, -3 - 1j], [-3 + 1j, -2]]) | |
| >>> E = mp.eighe(A, eigvals_only = True) | |
| >>> print(E) | |
| [-4.0] | |
| [ 3.0] | |
| >>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]]) | |
| >>> E, Q = mp.eighe(A) | |
| >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) | |
| [0.0] | |
| [0.0] | |
| see also: eigsy, eigh, eig | |
| """ | |
| if not overwrite_a: | |
| A = A.copy() | |
| d = ctx.zeros(A.rows, 1) | |
| e = ctx.zeros(A.rows, 1) | |
| t = ctx.zeros(A.rows, 1) | |
| if eigvals_only: | |
| c_he_tridiag_0(ctx, A, d, e, t) | |
| tridiag_eigen(ctx, d, e, False) | |
| return d | |
| else: | |
| c_he_tridiag_0(ctx, A, d, e, t) | |
| B = ctx.eye(A.rows) | |
| tridiag_eigen(ctx, d, e, B) | |
| c_he_tridiag_2(ctx, A, t, B) | |
| return (d, B) | |
| def eigh(ctx, A, eigvals_only = False, overwrite_a = False): | |
| """ | |
| "eigh" is a unified interface for "eigsy" and "eighe". Depending on | |
| whether A is real or complex the appropriate function is called. | |
| This routine solves the (ordinary) eigenvalue problem for a real symmetric | |
| or complex hermitian square matrix A. Given A, an orthogonal (A real) or | |
| unitary (A complex) matrix Q is calculated which diagonalizes A: | |
| Q' A Q = diag(E) and Q Q' = Q' Q = 1 | |
| Here diag(E) a is diagonal matrix whose diagonal is E. | |
| ' denotes the hermitian transpose (i.e. ordinary transposition and | |
| complex conjugation). | |
| The columns of Q are the eigenvectors of A and E contains the eigenvalues: | |
| A Q[:,i] = E[i] Q[:,i] | |
| input: | |
| A: a real or complex square matrix of format (n,n) which is symmetric | |
| (i.e. A[i,j]=A[j,i]) or hermitian (i.e. A[i,j]=conj(A[j,i])). | |
| eigvals_only: if true, calculates only the eigenvalues E. | |
| if false, calculates both eigenvectors and eigenvalues. | |
| overwrite_a: if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| E: vector of format (n). contains the eigenvalues of A in ascending order. | |
| Q: an orthogonal or unitary matrix of format (n,n). contains the | |
| eigenvectors of A as columns. | |
| return value: | |
| E if eigvals_only is true | |
| (E, Q) if eigvals_only is false | |
| example: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[3, 2], [2, 0]]) | |
| >>> E = mp.eigh(A, eigvals_only = True) | |
| >>> print(E) | |
| [-1.0] | |
| [ 4.0] | |
| >>> A = mp.matrix([[1, 2], [2, 3]]) | |
| >>> E, Q = mp.eigh(A) | |
| >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) | |
| [0.0] | |
| [0.0] | |
| >>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]]) | |
| >>> E, Q = mp.eigh(A) | |
| >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) | |
| [0.0] | |
| [0.0] | |
| see also: eigsy, eighe, eig | |
| """ | |
| iscomplex = any(type(x) is ctx.mpc for x in A) | |
| if iscomplex: | |
| return ctx.eighe(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a) | |
| else: | |
| return ctx.eigsy(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a) | |
| def gauss_quadrature(ctx, n, qtype = "legendre", alpha = 0, beta = 0): | |
| """ | |
| This routine calulates gaussian quadrature rules for different | |
| families of orthogonal polynomials. Let (a, b) be an interval, | |
| W(x) a positive weight function and n a positive integer. | |
| Then the purpose of this routine is to calculate pairs (x_k, w_k) | |
| for k=0, 1, 2, ... (n-1) which give | |
| int(W(x) * F(x), x = a..b) = sum(w_k * F(x_k),k = 0..(n-1)) | |
| exact for all polynomials F(x) of degree (strictly) less than 2*n. For all | |
| integrable functions F(x) the sum is a (more or less) good approximation to | |
| the integral. The x_k are called nodes (which are the zeros of the | |
| related orthogonal polynomials) and the w_k are called the weights. | |
| parameters | |
| n (input) The degree of the quadrature rule, i.e. its number of | |
| nodes. | |
| qtype (input) The family of orthogonal polynmomials for which to | |
| compute the quadrature rule. See the list below. | |
| alpha (input) real number, used as parameter for some orthogonal | |
| polynomials | |
| beta (input) real number, used as parameter for some orthogonal | |
| polynomials. | |
| return value | |
| (X, W) a pair of two real arrays where x_k = X[k] and w_k = W[k]. | |
| orthogonal polynomials: | |
| qtype polynomial | |
| ----- ---------- | |
| "legendre" Legendre polynomials, W(x)=1 on the interval (-1, +1) | |
| "legendre01" shifted Legendre polynomials, W(x)=1 on the interval (0, +1) | |
| "hermite" Hermite polynomials, W(x)=exp(-x*x) on (-infinity,+infinity) | |
| "laguerre" Laguerre polynomials, W(x)=exp(-x) on (0,+infinity) | |
| "glaguerre" generalized Laguerre polynomials, W(x)=exp(-x)*x**alpha | |
| on (0, +infinity) | |
| "chebyshev1" Chebyshev polynomials of the first kind, W(x)=1/sqrt(1-x*x) | |
| on (-1, +1) | |
| "chebyshev2" Chebyshev polynomials of the second kind, W(x)=sqrt(1-x*x) | |
| on (-1, +1) | |
| "jacobi" Jacobi polynomials, W(x)=(1-x)**alpha * (1+x)**beta on (-1, +1) | |
| with alpha>-1 and beta>-1 | |
| examples: | |
| >>> from mpmath import mp | |
| >>> f = lambda x: x**8 + 2 * x**6 - 3 * x**4 + 5 * x**2 - 7 | |
| >>> X, W = mp.gauss_quadrature(5, "hermite") | |
| >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) | |
| >>> B = mp.sqrt(mp.pi) * 57 / 16 | |
| >>> C = mp.quad(lambda x: mp.exp(- x * x) * f(x), [-mp.inf, +mp.inf]) | |
| >>> mp.nprint((mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10))) | |
| (0.0, 0.0) | |
| >>> f = lambda x: x**5 - 2 * x**4 + 3 * x**3 - 5 * x**2 + 7 * x - 11 | |
| >>> X, W = mp.gauss_quadrature(3, "laguerre") | |
| >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) | |
| >>> B = 76 | |
| >>> C = mp.quad(lambda x: mp.exp(-x) * f(x), [0, +mp.inf]) | |
| >>> mp.nprint(mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10)) | |
| .0 | |
| # orthogonality of the chebyshev polynomials: | |
| >>> f = lambda x: mp.chebyt(3, x) * mp.chebyt(2, x) | |
| >>> X, W = mp.gauss_quadrature(3, "chebyshev1") | |
| >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) | |
| >>> print(mp.chop(A, tol = 1e-10)) | |
| 0.0 | |
| references: | |
| - golub and welsch, "calculations of gaussian quadrature rules", mathematics of | |
| computation 23, p. 221-230 (1969) | |
| - golub, "some modified matrix eigenvalue problems", siam review 15, p. 318-334 (1973) | |
| - stroud and secrest, "gaussian quadrature formulas", prentice-hall (1966) | |
| See also the routine gaussq.f in netlog.org or ACM Transactions on | |
| Mathematical Software algorithm 726. | |
| """ | |
| d = ctx.zeros(n, 1) | |
| e = ctx.zeros(n, 1) | |
| z = ctx.zeros(1, n) | |
| z[0,0] = 1 | |
| if qtype == "legendre": | |
| # legendre on the range -1 +1 , abramowitz, table 25.4, p.916 | |
| w = 2 | |
| for i in xrange(n): | |
| j = i + 1 | |
| e[i] = ctx.sqrt(j * j / (4 * j * j - ctx.mpf(1))) | |
| elif qtype == "legendre01": | |
| # legendre shifted to 0 1 , abramowitz, table 25.8, p.921 | |
| w = 1 | |
| for i in xrange(n): | |
| d[i] = 1 / ctx.mpf(2) | |
| j = i + 1 | |
| e[i] = ctx.sqrt(j * j / (16 * j * j - ctx.mpf(4))) | |
| elif qtype == "hermite": | |
| # hermite on the range -inf +inf , abramowitz, table 25.10,p.924 | |
| w = ctx.sqrt(ctx.pi) | |
| for i in xrange(n): | |
| j = i + 1 | |
| e[i] = ctx.sqrt(j / ctx.mpf(2)) | |
| elif qtype == "laguerre": | |
| # laguerre on the range 0 +inf , abramowitz, table 25.9, p. 923 | |
| w = 1 | |
| for i in xrange(n): | |
| j = i + 1 | |
| d[i] = 2 * j - 1 | |
| e[i] = j | |
| elif qtype=="chebyshev1": | |
| # chebyshev polynimials of the first kind | |
| w = ctx.pi | |
| for i in xrange(n): | |
| e[i] = 1 / ctx.mpf(2) | |
| e[0] = ctx.sqrt(1 / ctx.mpf(2)) | |
| elif qtype == "chebyshev2": | |
| # chebyshev polynimials of the second kind | |
| w = ctx.pi / 2 | |
| for i in xrange(n): | |
| e[i] = 1 / ctx.mpf(2) | |
| elif qtype == "glaguerre": | |
| # generalized laguerre on the range 0 +inf | |
| w = ctx.gamma(1 + alpha) | |
| for i in xrange(n): | |
| j = i + 1 | |
| d[i] = 2 * j - 1 + alpha | |
| e[i] = ctx.sqrt(j * (j + alpha)) | |
| elif qtype == "jacobi": | |
| # jacobi polynomials | |
| alpha = ctx.mpf(alpha) | |
| beta = ctx.mpf(beta) | |
| ab = alpha + beta | |
| abi = ab + 2 | |
| w = (2**(ab+1)) * ctx.gamma(alpha + 1) * ctx.gamma(beta + 1) / ctx.gamma(abi) | |
| d[0] = (beta - alpha) / abi | |
| e[0] = ctx.sqrt(4 * (1 + alpha) * (1 + beta) / ((abi + 1) * (abi * abi))) | |
| a2b2 = beta * beta - alpha * alpha | |
| for i in xrange(1, n): | |
| j = i + 1 | |
| abi = 2 * j + ab | |
| d[i] = a2b2 / ((abi - 2) * abi) | |
| e[i] = ctx.sqrt(4 * j * (j + alpha) * (j + beta) * (j + ab) / ((abi * abi - 1) * abi * abi)) | |
| elif isinstance(qtype, str): | |
| raise ValueError("unknown quadrature rule \"%s\"" % qtype) | |
| elif not isinstance(qtype, str): | |
| w = qtype(d, e) | |
| else: | |
| assert 0 | |
| tridiag_eigen(ctx, d, e, z) | |
| for i in xrange(len(z)): | |
| z[i] *= z[i] | |
| z = z.transpose() | |
| return (d, w * z) | |
| ################################################################################################## | |
| ################################################################################################## | |
| ################################################################################################## | |
| def svd_r_raw(ctx, A, V = False, calc_u = False): | |
| """ | |
| This routine computes the singular value decomposition of a matrix A. | |
| Given A, two orthogonal matrices U and V are calculated such that | |
| A = U S V | |
| where S is a suitable shaped matrix whose off-diagonal elements are zero. | |
| The diagonal elements of S are the singular values of A, i.e. the | |
| squareroots of the eigenvalues of A' A or A A'. Here ' denotes the transpose. | |
| Householder bidiagonalization and a variant of the QR algorithm is used. | |
| overview of the matrices : | |
| A : m*n A gets replaced by U | |
| U : m*n U replaces A. If n>m then only the first m*m block of U is | |
| non-zero. column-orthogonal: U' U = B | |
| here B is a n*n matrix whose first min(m,n) diagonal | |
| elements are 1 and all other elements are zero. | |
| S : n*n diagonal matrix, only the diagonal elements are stored in | |
| the array S. only the first min(m,n) diagonal elements are non-zero. | |
| V : n*n orthogonal: V V' = V' V = 1 | |
| parameters: | |
| A (input/output) On input, A contains a real matrix of shape m*n. | |
| On output, if calc_u is true A contains the column-orthogonal | |
| matrix U; otherwise A is simply used as workspace and thus destroyed. | |
| V (input/output) if false, the matrix V is not calculated. otherwise | |
| V must be a matrix of shape n*n. | |
| calc_u (input) If true, the matrix U is calculated and replaces A. | |
| if false, U is not calculated and A is simply destroyed | |
| return value: | |
| S an array of length n containing the singular values of A sorted by | |
| decreasing magnitude. only the first min(m,n) elements are non-zero. | |
| This routine is a python translation of the fortran routine svd.f in the | |
| software library EISPACK (see netlib.org) which itself is based on the | |
| algol procedure svd described in: | |
| - num. math. 14, 403-420(1970) by golub and reinsch. | |
| - wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971). | |
| """ | |
| m, n = A.rows, A.cols | |
| S = ctx.zeros(n, 1) | |
| # work is a temporary array of size n | |
| work = ctx.zeros(n, 1) | |
| g = scale = anorm = 0 | |
| maxits = 3 * ctx.dps | |
| for i in xrange(n): # householder reduction to bidiagonal form | |
| work[i] = scale*g | |
| g = s = scale = 0 | |
| if i < m: | |
| for k in xrange(i, m): | |
| scale += ctx.fabs(A[k,i]) | |
| if scale != 0: | |
| for k in xrange(i, m): | |
| A[k,i] /= scale | |
| s += A[k,i] * A[k,i] | |
| f = A[i,i] | |
| g = -ctx.sqrt(s) | |
| if f < 0: | |
| g = -g | |
| h = f * g - s | |
| A[i,i] = f - g | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i, m): | |
| s += A[k,i] * A[k,j] | |
| f = s / h | |
| for k in xrange(i, m): | |
| A[k,j] += f * A[k,i] | |
| for k in xrange(i,m): | |
| A[k,i] *= scale | |
| S[i] = scale * g | |
| g = s = scale = 0 | |
| if i < m and i != n - 1: | |
| for k in xrange(i+1, n): | |
| scale += ctx.fabs(A[i,k]) | |
| if scale: | |
| for k in xrange(i+1, n): | |
| A[i,k] /= scale | |
| s += A[i,k] * A[i,k] | |
| f = A[i,i+1] | |
| g = -ctx.sqrt(s) | |
| if f < 0: | |
| g = -g | |
| h = f * g - s | |
| A[i,i+1] = f - g | |
| for k in xrange(i+1, n): | |
| work[k] = A[i,k] / h | |
| for j in xrange(i+1, m): | |
| s = 0 | |
| for k in xrange(i+1, n): | |
| s += A[j,k] * A[i,k] | |
| for k in xrange(i+1, n): | |
| A[j,k] += s * work[k] | |
| for k in xrange(i+1, n): | |
| A[i,k] *= scale | |
| anorm = max(anorm, ctx.fabs(S[i]) + ctx.fabs(work[i])) | |
| if not isinstance(V, bool): | |
| for i in xrange(n-2, -1, -1): # accumulation of right hand transformations | |
| V[i+1,i+1] = 1 | |
| if work[i+1] != 0: | |
| for j in xrange(i+1, n): | |
| V[i,j] = (A[i,j] / A[i,i+1]) / work[i+1] | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i+1, n): | |
| s += A[i,k] * V[j,k] | |
| for k in xrange(i+1, n): | |
| V[j,k] += s * V[i,k] | |
| for j in xrange(i+1, n): | |
| V[j,i] = V[i,j] = 0 | |
| V[0,0] = 1 | |
| if m<n : minnm = m | |
| else : minnm = n | |
| if calc_u: | |
| for i in xrange(minnm-1, -1, -1): # accumulation of left hand transformations | |
| g = S[i] | |
| for j in xrange(i+1, n): | |
| A[i,j] = 0 | |
| if g != 0: | |
| g = 1 / g | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i+1, m): | |
| s += A[k,i] * A[k,j] | |
| f = (s / A[i,i]) * g | |
| for k in xrange(i, m): | |
| A[k,j] += f * A[k,i] | |
| for j in xrange(i, m): | |
| A[j,i] *= g | |
| else: | |
| for j in xrange(i, m): | |
| A[j,i] = 0 | |
| A[i,i] += 1 | |
| for k in xrange(n - 1, -1, -1): | |
| # diagonalization of the bidiagonal form: | |
| # loop over singular values, and over allowed itations | |
| its = 0 | |
| while 1: | |
| its += 1 | |
| flag = True | |
| for l in xrange(k, -1, -1): | |
| nm = l-1 | |
| if ctx.fabs(work[l]) + anorm == anorm: | |
| flag = False | |
| break | |
| if ctx.fabs(S[nm]) + anorm == anorm: | |
| break | |
| if flag: | |
| c = 0 | |
| s = 1 | |
| for i in xrange(l, k + 1): | |
| f = s * work[i] | |
| work[i] *= c | |
| if ctx.fabs(f) + anorm == anorm: | |
| break | |
| g = S[i] | |
| h = ctx.hypot(f, g) | |
| S[i] = h | |
| h = 1 / h | |
| c = g * h | |
| s = - f * h | |
| if calc_u: | |
| for j in xrange(m): | |
| y = A[j,nm] | |
| z = A[j,i] | |
| A[j,nm] = y * c + z * s | |
| A[j,i] = z * c - y * s | |
| z = S[k] | |
| if l == k: # convergence | |
| if z < 0: # singular value is made nonnegative | |
| S[k] = -z | |
| if not isinstance(V, bool): | |
| for j in xrange(n): | |
| V[k,j] = -V[k,j] | |
| break | |
| if its >= maxits: | |
| raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its) | |
| x = S[l] # shift from bottom 2 by 2 minor | |
| nm = k-1 | |
| y = S[nm] | |
| g = work[nm] | |
| h = work[k] | |
| f = ((y - z) * (y + z) + (g - h) * (g + h))/(2 * h * y) | |
| g = ctx.hypot(f, 1) | |
| if f >= 0: f = ((x - z) * (x + z) + h * ((y / (f + g)) - h)) / x | |
| else: f = ((x - z) * (x + z) + h * ((y / (f - g)) - h)) / x | |
| c = s = 1 # next qt transformation | |
| for j in xrange(l, nm + 1): | |
| g = work[j+1] | |
| y = S[j+1] | |
| h = s * g | |
| g = c * g | |
| z = ctx.hypot(f, h) | |
| work[j] = z | |
| c = f / z | |
| s = h / z | |
| f = x * c + g * s | |
| g = g * c - x * s | |
| h = y * s | |
| y *= c | |
| if not isinstance(V, bool): | |
| for jj in xrange(n): | |
| x = V[j ,jj] | |
| z = V[j+1,jj] | |
| V[j ,jj]= x * c + z * s | |
| V[j+1 ,jj]= z * c - x * s | |
| z = ctx.hypot(f, h) | |
| S[j] = z | |
| if z != 0: # rotation can be arbitray if z=0 | |
| z = 1 / z | |
| c = f * z | |
| s = h * z | |
| f = c * g + s * y | |
| x = c * y - s * g | |
| if calc_u: | |
| for jj in xrange(m): | |
| y = A[jj,j ] | |
| z = A[jj,j+1] | |
| A[jj,j ] = y * c + z * s | |
| A[jj,j+1 ] = z * c - y * s | |
| work[l] = 0 | |
| work[k] = f | |
| S[k] = x | |
| ########################## | |
| # Sort singular values into decreasing order (bubble-sort) | |
| for i in xrange(n): | |
| imax = i | |
| s = ctx.fabs(S[i]) # s is the current maximal element | |
| for j in xrange(i + 1, n): | |
| c = ctx.fabs(S[j]) | |
| if c > s: | |
| s = c | |
| imax = j | |
| if imax != i: | |
| # swap singular values | |
| z = S[i] | |
| S[i] = S[imax] | |
| S[imax] = z | |
| if calc_u: | |
| for j in xrange(m): | |
| z = A[j,i] | |
| A[j,i] = A[j,imax] | |
| A[j,imax] = z | |
| if not isinstance(V, bool): | |
| for j in xrange(n): | |
| z = V[i,j] | |
| V[i,j] = V[imax,j] | |
| V[imax,j] = z | |
| return S | |
| ####################### | |
| def svd_c_raw(ctx, A, V = False, calc_u = False): | |
| """ | |
| This routine computes the singular value decomposition of a matrix A. | |
| Given A, two unitary matrices U and V are calculated such that | |
| A = U S V | |
| where S is a suitable shaped matrix whose off-diagonal elements are zero. | |
| The diagonal elements of S are the singular values of A, i.e. the | |
| squareroots of the eigenvalues of A' A or A A'. Here ' denotes the hermitian | |
| transpose (i.e. transposition and conjugation). Householder bidiagonalization | |
| and a variant of the QR algorithm is used. | |
| overview of the matrices : | |
| A : m*n A gets replaced by U | |
| U : m*n U replaces A. If n>m then only the first m*m block of U is | |
| non-zero. column-unitary: U' U = B | |
| here B is a n*n matrix whose first min(m,n) diagonal | |
| elements are 1 and all other elements are zero. | |
| S : n*n diagonal matrix, only the diagonal elements are stored in | |
| the array S. only the first min(m,n) diagonal elements are non-zero. | |
| V : n*n unitary: V V' = V' V = 1 | |
| parameters: | |
| A (input/output) On input, A contains a complex matrix of shape m*n. | |
| On output, if calc_u is true A contains the column-unitary | |
| matrix U; otherwise A is simply used as workspace and thus destroyed. | |
| V (input/output) if false, the matrix V is not calculated. otherwise | |
| V must be a matrix of shape n*n. | |
| calc_u (input) If true, the matrix U is calculated and replaces A. | |
| if false, U is not calculated and A is simply destroyed | |
| return value: | |
| S an array of length n containing the singular values of A sorted by | |
| decreasing magnitude. only the first min(m,n) elements are non-zero. | |
| This routine is a python translation of the fortran routine svd.f in the | |
| software library EISPACK (see netlib.org) which itself is based on the | |
| algol procedure svd described in: | |
| - num. math. 14, 403-420(1970) by golub and reinsch. | |
| - wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971). | |
| """ | |
| m, n = A.rows, A.cols | |
| S = ctx.zeros(n, 1) | |
| # work is a temporary array of size n | |
| work = ctx.zeros(n, 1) | |
| lbeta = ctx.zeros(n, 1) | |
| rbeta = ctx.zeros(n, 1) | |
| dwork = ctx.zeros(n, 1) | |
| g = scale = anorm = 0 | |
| maxits = 3 * ctx.dps | |
| for i in xrange(n): # householder reduction to bidiagonal form | |
| dwork[i] = scale * g # dwork are the side-diagonal elements | |
| g = s = scale = 0 | |
| if i < m: | |
| for k in xrange(i, m): | |
| scale += ctx.fabs(ctx.re(A[k,i])) + ctx.fabs(ctx.im(A[k,i])) | |
| if scale != 0: | |
| for k in xrange(i, m): | |
| A[k,i] /= scale | |
| ar = ctx.re(A[k,i]) | |
| ai = ctx.im(A[k,i]) | |
| s += ar * ar + ai * ai | |
| f = A[i,i] | |
| g = -ctx.sqrt(s) | |
| if ctx.re(f) < 0: | |
| beta = -g - ctx.conj(f) | |
| g = -g | |
| else: | |
| beta = -g + ctx.conj(f) | |
| beta /= ctx.conj(beta) | |
| beta += 1 | |
| h = 2 * (ctx.re(f) * g - s) | |
| A[i,i] = f - g | |
| beta /= h | |
| lbeta[i] = (beta / scale) / scale | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i, m): | |
| s += ctx.conj(A[k,i]) * A[k,j] | |
| f = beta * s | |
| for k in xrange(i, m): | |
| A[k,j] += f * A[k,i] | |
| for k in xrange(i, m): | |
| A[k,i] *= scale | |
| S[i] = scale * g # S are the diagonal elements | |
| g = s = scale = 0 | |
| if i < m and i != n - 1: | |
| for k in xrange(i+1, n): | |
| scale += ctx.fabs(ctx.re(A[i,k])) + ctx.fabs(ctx.im(A[i,k])) | |
| if scale: | |
| for k in xrange(i+1, n): | |
| A[i,k] /= scale | |
| ar = ctx.re(A[i,k]) | |
| ai = ctx.im(A[i,k]) | |
| s += ar * ar + ai * ai | |
| f = A[i,i+1] | |
| g = -ctx.sqrt(s) | |
| if ctx.re(f) < 0: | |
| beta = -g - ctx.conj(f) | |
| g = -g | |
| else: | |
| beta = -g + ctx.conj(f) | |
| beta /= ctx.conj(beta) | |
| beta += 1 | |
| h = 2 * (ctx.re(f) * g - s) | |
| A[i,i+1] = f - g | |
| beta /= h | |
| rbeta[i] = (beta / scale) / scale | |
| for k in xrange(i+1, n): | |
| work[k] = A[i, k] | |
| for j in xrange(i+1, m): | |
| s = 0 | |
| for k in xrange(i+1, n): | |
| s += ctx.conj(A[i,k]) * A[j,k] | |
| f = s * beta | |
| for k in xrange(i+1,n): | |
| A[j,k] += f * work[k] | |
| for k in xrange(i+1, n): | |
| A[i,k] *= scale | |
| anorm = max(anorm,ctx.fabs(S[i]) + ctx.fabs(dwork[i])) | |
| if not isinstance(V, bool): | |
| for i in xrange(n-2, -1, -1): # accumulation of right hand transformations | |
| V[i+1,i+1] = 1 | |
| if dwork[i+1] != 0: | |
| f = ctx.conj(rbeta[i]) | |
| for j in xrange(i+1, n): | |
| V[i,j] = A[i,j] * f | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i+1, n): | |
| s += ctx.conj(A[i,k]) * V[j,k] | |
| for k in xrange(i+1, n): | |
| V[j,k] += s * V[i,k] | |
| for j in xrange(i+1,n): | |
| V[j,i] = V[i,j] = 0 | |
| V[0,0] = 1 | |
| if m < n : minnm = m | |
| else : minnm = n | |
| if calc_u: | |
| for i in xrange(minnm-1, -1, -1): # accumulation of left hand transformations | |
| g = S[i] | |
| for j in xrange(i+1, n): | |
| A[i,j] = 0 | |
| if g != 0: | |
| g = 1 / g | |
| for j in xrange(i+1, n): | |
| s = 0 | |
| for k in xrange(i+1, m): | |
| s += ctx.conj(A[k,i]) * A[k,j] | |
| f = s * ctx.conj(lbeta[i]) | |
| for k in xrange(i, m): | |
| A[k,j] += f * A[k,i] | |
| for j in xrange(i, m): | |
| A[j,i] *= g | |
| else: | |
| for j in xrange(i, m): | |
| A[j,i] = 0 | |
| A[i,i] += 1 | |
| for k in xrange(n-1, -1, -1): | |
| # diagonalization of the bidiagonal form: | |
| # loop over singular values, and over allowed itations | |
| its = 0 | |
| while 1: | |
| its += 1 | |
| flag = True | |
| for l in xrange(k, -1, -1): | |
| nm = l - 1 | |
| if ctx.fabs(dwork[l]) + anorm == anorm: | |
| flag = False | |
| break | |
| if ctx.fabs(S[nm]) + anorm == anorm: | |
| break | |
| if flag: | |
| c = 0 | |
| s = 1 | |
| for i in xrange(l, k+1): | |
| f = s * dwork[i] | |
| dwork[i] *= c | |
| if ctx.fabs(f) + anorm == anorm: | |
| break | |
| g = S[i] | |
| h = ctx.hypot(f, g) | |
| S[i] = h | |
| h = 1 / h | |
| c = g * h | |
| s = -f * h | |
| if calc_u: | |
| for j in xrange(m): | |
| y = A[j,nm] | |
| z = A[j,i] | |
| A[j,nm]= y * c + z * s | |
| A[j,i] = z * c - y * s | |
| z = S[k] | |
| if l == k: # convergence | |
| if z < 0: # singular value is made nonnegative | |
| S[k] = -z | |
| if not isinstance(V, bool): | |
| for j in xrange(n): | |
| V[k,j] = -V[k,j] | |
| break | |
| if its >= maxits: | |
| raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its) | |
| x = S[l] # shift from bottom 2 by 2 minor | |
| nm = k-1 | |
| y = S[nm] | |
| g = dwork[nm] | |
| h = dwork[k] | |
| f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2 * h * y) | |
| g = ctx.hypot(f, 1) | |
| if f >=0: f = (( x - z) *( x + z) + h *((y / (f + g)) - h)) / x | |
| else: f = (( x - z) *( x + z) + h *((y / (f - g)) - h)) / x | |
| c = s = 1 # next qt transformation | |
| for j in xrange(l, nm + 1): | |
| g = dwork[j+1] | |
| y = S[j+1] | |
| h = s * g | |
| g = c * g | |
| z = ctx.hypot(f, h) | |
| dwork[j] = z | |
| c = f / z | |
| s = h / z | |
| f = x * c + g * s | |
| g = g * c - x * s | |
| h = y * s | |
| y *= c | |
| if not isinstance(V, bool): | |
| for jj in xrange(n): | |
| x = V[j ,jj] | |
| z = V[j+1,jj] | |
| V[j ,jj]= x * c + z * s | |
| V[j+1,jj ]= z * c - x * s | |
| z = ctx.hypot(f, h) | |
| S[j] = z | |
| if z != 0: # rotation can be arbitray if z=0 | |
| z = 1 / z | |
| c = f * z | |
| s = h * z | |
| f = c * g + s * y | |
| x = c * y - s * g | |
| if calc_u: | |
| for jj in xrange(m): | |
| y = A[jj,j ] | |
| z = A[jj,j+1] | |
| A[jj,j ]= y * c + z * s | |
| A[jj,j+1 ]= z * c - y * s | |
| dwork[l] = 0 | |
| dwork[k] = f | |
| S[k] = x | |
| ########################## | |
| # Sort singular values into decreasing order (bubble-sort) | |
| for i in xrange(n): | |
| imax = i | |
| s = ctx.fabs(S[i]) # s is the current maximal element | |
| for j in xrange(i + 1, n): | |
| c = ctx.fabs(S[j]) | |
| if c > s: | |
| s = c | |
| imax = j | |
| if imax != i: | |
| # swap singular values | |
| z = S[i] | |
| S[i] = S[imax] | |
| S[imax] = z | |
| if calc_u: | |
| for j in xrange(m): | |
| z = A[j,i] | |
| A[j,i] = A[j,imax] | |
| A[j,imax] = z | |
| if not isinstance(V, bool): | |
| for j in xrange(n): | |
| z = V[i,j] | |
| V[i,j] = V[imax,j] | |
| V[imax,j] = z | |
| return S | |
| ################################################################################################## | |
| def svd_r(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): | |
| """ | |
| This routine computes the singular value decomposition of a matrix A. | |
| Given A, two orthogonal matrices U and V are calculated such that | |
| A = U S V and U' U = 1 and V V' = 1 | |
| where S is a suitable shaped matrix whose off-diagonal elements are zero. | |
| Here ' denotes the transpose. The diagonal elements of S are the singular | |
| values of A, i.e. the squareroots of the eigenvalues of A' A or A A'. | |
| input: | |
| A : a real matrix of shape (m, n) | |
| full_matrices : if true, U and V are of shape (m, m) and (n, n). | |
| if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). | |
| compute_uv : if true, U and V are calculated. if false, only S is calculated. | |
| overwrite_a : if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| U : an orthogonal matrix: U' U = 1. if full_matrices is true, U is of | |
| shape (m, m). ortherwise it is of shape (m, min(m, n)). | |
| S : an array of length min(m, n) containing the singular values of A sorted by | |
| decreasing magnitude. | |
| V : an orthogonal matrix: V V' = 1. if full_matrices is true, V is of | |
| shape (n, n). ortherwise it is of shape (min(m, n), n). | |
| return value: | |
| S if compute_uv is false | |
| (U, S, V) if compute_uv is true | |
| overview of the matrices: | |
| full_matrices true: | |
| A : m*n | |
| U : m*m U' U = 1 | |
| S as matrix : m*n | |
| V : n*n V V' = 1 | |
| full_matrices false: | |
| A : m*n | |
| U : m*min(n,m) U' U = 1 | |
| S as matrix : min(m,n)*min(m,n) | |
| V : min(m,n)*n V V' = 1 | |
| examples: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]]) | |
| >>> S = mp.svd_r(A, compute_uv = False) | |
| >>> print(S) | |
| [6.0] | |
| [3.0] | |
| [1.0] | |
| >>> U, S, V = mp.svd_r(A) | |
| >>> print(mp.chop(A - U * mp.diag(S) * V)) | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| see also: svd, svd_c | |
| """ | |
| m, n = A.rows, A.cols | |
| if not compute_uv: | |
| if not overwrite_a: | |
| A = A.copy() | |
| S = svd_r_raw(ctx, A, V = False, calc_u = False) | |
| S = S[:min(m,n)] | |
| return S | |
| if full_matrices and n < m: | |
| V = ctx.zeros(m, m) | |
| A0 = ctx.zeros(m, m) | |
| A0[:,:n] = A | |
| S = svd_r_raw(ctx, A0, V, calc_u = True) | |
| S = S[:n] | |
| V = V[:n,:n] | |
| return (A0, S, V) | |
| else: | |
| if not overwrite_a: | |
| A = A.copy() | |
| V = ctx.zeros(n, n) | |
| S = svd_r_raw(ctx, A, V, calc_u = True) | |
| if n > m: | |
| if full_matrices == False: | |
| V = V[:m,:] | |
| S = S[:m] | |
| A = A[:,:m] | |
| return (A, S, V) | |
| ############################## | |
| def svd_c(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): | |
| """ | |
| This routine computes the singular value decomposition of a matrix A. | |
| Given A, two unitary matrices U and V are calculated such that | |
| A = U S V and U' U = 1 and V V' = 1 | |
| where S is a suitable shaped matrix whose off-diagonal elements are zero. | |
| Here ' denotes the hermitian transpose (i.e. transposition and complex | |
| conjugation). The diagonal elements of S are the singular values of A, | |
| i.e. the squareroots of the eigenvalues of A' A or A A'. | |
| input: | |
| A : a complex matrix of shape (m, n) | |
| full_matrices : if true, U and V are of shape (m, m) and (n, n). | |
| if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). | |
| compute_uv : if true, U and V are calculated. if false, only S is calculated. | |
| overwrite_a : if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| U : an unitary matrix: U' U = 1. if full_matrices is true, U is of | |
| shape (m, m). ortherwise it is of shape (m, min(m, n)). | |
| S : an array of length min(m, n) containing the singular values of A sorted by | |
| decreasing magnitude. | |
| V : an unitary matrix: V V' = 1. if full_matrices is true, V is of | |
| shape (n, n). ortherwise it is of shape (min(m, n), n). | |
| return value: | |
| S if compute_uv is false | |
| (U, S, V) if compute_uv is true | |
| overview of the matrices: | |
| full_matrices true: | |
| A : m*n | |
| U : m*m U' U = 1 | |
| S as matrix : m*n | |
| V : n*n V V' = 1 | |
| full_matrices false: | |
| A : m*n | |
| U : m*min(n,m) U' U = 1 | |
| S as matrix : min(m,n)*min(m,n) | |
| V : min(m,n)*n V V' = 1 | |
| example: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[-2j, -1-3j, -2+2j], [2-2j, -1-3j, 1], [-3+1j,-2j,0]]) | |
| >>> S = mp.svd_c(A, compute_uv = False) | |
| >>> print(mp.chop(S - mp.matrix([mp.sqrt(34), mp.sqrt(15), mp.sqrt(6)]))) | |
| [0.0] | |
| [0.0] | |
| [0.0] | |
| >>> U, S, V = mp.svd_c(A) | |
| >>> print(mp.chop(A - U * mp.diag(S) * V)) | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| see also: svd, svd_r | |
| """ | |
| m, n = A.rows, A.cols | |
| if not compute_uv: | |
| if not overwrite_a: | |
| A = A.copy() | |
| S = svd_c_raw(ctx, A, V = False, calc_u = False) | |
| S = S[:min(m,n)] | |
| return S | |
| if full_matrices and n < m: | |
| V = ctx.zeros(m, m) | |
| A0 = ctx.zeros(m, m) | |
| A0[:,:n] = A | |
| S = svd_c_raw(ctx, A0, V, calc_u = True) | |
| S = S[:n] | |
| V = V[:n,:n] | |
| return (A0, S, V) | |
| else: | |
| if not overwrite_a: | |
| A = A.copy() | |
| V = ctx.zeros(n, n) | |
| S = svd_c_raw(ctx, A, V, calc_u = True) | |
| if n > m: | |
| if full_matrices == False: | |
| V = V[:m,:] | |
| S = S[:m] | |
| A = A[:,:m] | |
| return (A, S, V) | |
| def svd(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): | |
| """ | |
| "svd" is a unified interface for "svd_r" and "svd_c". Depending on | |
| whether A is real or complex the appropriate function is called. | |
| This routine computes the singular value decomposition of a matrix A. | |
| Given A, two orthogonal (A real) or unitary (A complex) matrices U and V | |
| are calculated such that | |
| A = U S V and U' U = 1 and V V' = 1 | |
| where S is a suitable shaped matrix whose off-diagonal elements are zero. | |
| Here ' denotes the hermitian transpose (i.e. transposition and complex | |
| conjugation). The diagonal elements of S are the singular values of A, | |
| i.e. the squareroots of the eigenvalues of A' A or A A'. | |
| input: | |
| A : a real or complex matrix of shape (m, n) | |
| full_matrices : if true, U and V are of shape (m, m) and (n, n). | |
| if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). | |
| compute_uv : if true, U and V are calculated. if false, only S is calculated. | |
| overwrite_a : if true, allows modification of A which may improve | |
| performance. if false, A is not modified. | |
| output: | |
| U : an orthogonal or unitary matrix: U' U = 1. if full_matrices is true, U is of | |
| shape (m, m). ortherwise it is of shape (m, min(m, n)). | |
| S : an array of length min(m, n) containing the singular values of A sorted by | |
| decreasing magnitude. | |
| V : an orthogonal or unitary matrix: V V' = 1. if full_matrices is true, V is of | |
| shape (n, n). ortherwise it is of shape (min(m, n), n). | |
| return value: | |
| S if compute_uv is false | |
| (U, S, V) if compute_uv is true | |
| overview of the matrices: | |
| full_matrices true: | |
| A : m*n | |
| U : m*m U' U = 1 | |
| S as matrix : m*n | |
| V : n*n V V' = 1 | |
| full_matrices false: | |
| A : m*n | |
| U : m*min(n,m) U' U = 1 | |
| S as matrix : min(m,n)*min(m,n) | |
| V : min(m,n)*n V V' = 1 | |
| examples: | |
| >>> from mpmath import mp | |
| >>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]]) | |
| >>> S = mp.svd(A, compute_uv = False) | |
| >>> print(S) | |
| [6.0] | |
| [3.0] | |
| [1.0] | |
| >>> U, S, V = mp.svd(A) | |
| >>> print(mp.chop(A - U * mp.diag(S) * V)) | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| [0.0 0.0 0.0] | |
| see also: svd_r, svd_c | |
| """ | |
| iscomplex = any(type(x) is ctx.mpc for x in A) | |
| if iscomplex: | |
| return ctx.svd_c(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a) | |
| else: | |
| return ctx.svd_r(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a) | |