#!/usr/bin/python # -*- coding: utf-8 -*- ################################################################################################## # module for the eigenvalue problem # Copyright 2013 Timo Hartmann (thartmann15 at gmail.com) # # todo: # - implement balancing # - agressive early deflation # ################################################################################################## """ The eigenvalue problem ---------------------- This file contains routines for the eigenvalue problem. high level routines: hessenberg : reduction of a real or complex square matrix to upper Hessenberg form schur : reduction of a real or complex square matrix to upper Schur form eig : eigenvalues and eigenvectors of a real or complex square matrix low level routines: hessenberg_reduce_0 : reduction of a real or complex square matrix to upper Hessenberg form hessenberg_reduce_1 : auxiliary routine to hessenberg_reduce_0 qr_step : a single implicitly shifted QR step for an upper Hessenberg matrix hessenberg_qr : Schur decomposition of an upper Hessenberg matrix eig_tr_r : right eigenvectors of an upper triangular matrix eig_tr_l : left eigenvectors of an upper triangular matrix """ from ..libmp.backend import xrange class Eigen(object): pass def defun(f): setattr(Eigen, f.__name__, f) return f def hessenberg_reduce_0(ctx, A, T): """ This routine computes the (upper) Hessenberg decomposition of a square matrix A. Given A, an unitary matrix Q is calculated such that Q' A Q = H and Q' Q = Q Q' = 1 where H is an upper Hessenberg matrix, meaning that it only contains zeros below the first subdiagonal. Here ' denotes the hermitian transpose (i.e. transposition and conjugation). parameters: A (input/output) On input, A contains the square matrix A of dimension (n,n). On output, A contains a compressed representation of Q and H. T (output) An array of length n containing the first elements of the Householder reflectors. """ # internally we work with householder reflections from the right. # let u be a row vector (i.e. u[i]=A[i,:i]). then # Q is build up by reflectors of the type (1-v'v) where v is a suitable # modification of u. these reflectors are applyed to A from the right. # because we work with reflectors from the right we have to start with # the bottom row of A and work then upwards (this corresponds to # some kind of RQ decomposition). # the first part of the vectors v (i.e. A[i,:(i-1)]) are stored as row vectors # in the lower left part of A (excluding the diagonal and subdiagonal). # the last entry of v is stored in T. # the upper right part of A (including diagonal and subdiagonal) becomes H. n = A.rows if n <= 2: return for i in xrange(n-1, 1, -1): # scale the vector scale = 0 for k in xrange(0, i): scale += abs(ctx.re(A[i,k])) + abs(ctx.im(A[i,k])) scale_inv = 0 if scale != 0: scale_inv = 1 / scale if scale == 0 or ctx.isinf(scale_inv): # sadly there are floating point numbers not equal to zero whose reciprocal is infinity T[i] = 0 A[i,i-1] = 0 continue # calculate parameters for housholder transformation H = 0 for k in xrange(0, i): A[i,k] *= scale_inv rr = ctx.re(A[i,k]) ii = ctx.im(A[i,k]) H += rr * rr + ii * ii F = A[i,i-1] f = abs(F) G = ctx.sqrt(H) A[i,i-1] = - G * scale if f == 0: T[i] = G else: ff = F / f T[i] = F + G * ff A[i,i-1] *= ff H += G * f H = 1 / ctx.sqrt(H) T[i] *= H for k in xrange(0, i - 1): A[i,k] *= H for j in xrange(0, i): # apply housholder transformation (from right) G = ctx.conj(T[i]) * A[j,i-1] for k in xrange(0, i-1): G += ctx.conj(A[i,k]) * A[j,k] A[j,i-1] -= G * T[i] for k in xrange(0, i-1): A[j,k] -= G * A[i,k] for j in xrange(0, n): # apply housholder transformation (from left) G = T[i] * A[i-1,j] for k in xrange(0, i-1): G += A[i,k] * A[k,j] A[i-1,j] -= G * ctx.conj(T[i]) for k in xrange(0, i-1): A[k,j] -= G * ctx.conj(A[i,k]) def hessenberg_reduce_1(ctx, A, T): """ This routine forms the unitary matrix Q described in hessenberg_reduce_0. parameters: A (input/output) On input, A is the same matrix as delivered by hessenberg_reduce_0. On output, A is set to Q. T (input) On input, T is the same array as delivered by hessenberg_reduce_0. """ n = A.rows if n == 1: A[0,0] = 1 return A[0,0] = A[1,1] = 1 A[0,1] = A[1,0] = 0 for i in xrange(2, n): if T[i] != 0: for j in xrange(0, i): G = T[i] * A[i-1,j] for k in xrange(0, i-1): G += A[i,k] * A[k,j] A[i-1,j] -= G * ctx.conj(T[i]) for k in xrange(0, i-1): A[k,j] -= G * ctx.conj(A[i,k]) A[i,i] = 1 for j in xrange(0, i): A[j,i] = A[i,j] = 0 @defun def hessenberg(ctx, A, overwrite_a = False): """ This routine computes the Hessenberg decomposition of a square matrix A. Given A, an unitary matrix Q is determined such that Q' A Q = H and Q' Q = Q Q' = 1 where H is an upper right Hessenberg matrix. Here ' denotes the hermitian transpose (i.e. transposition and conjugation). input: A : a real or complex square matrix overwrite_a : if true, allows modification of A which may improve performance. if false, A is not modified. output: Q : an unitary matrix H : an upper right Hessenberg matrix example: >>> from mpmath import mp >>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]]) >>> Q, H = mp.hessenberg(A) >>> mp.nprint(H, 3) # doctest:+SKIP [ 3.15 2.23 4.44] [-0.769 4.85 3.05] [ 0.0 3.61 7.0] >>> print(mp.chop(A - Q * H * Q.transpose_conj())) [0.0 0.0 0.0] [0.0 0.0 0.0] [0.0 0.0 0.0] return value: (Q, H) """ n = A.rows if n == 1: return (ctx.matrix([[1]]), A) if not overwrite_a: A = A.copy() T = ctx.matrix(n, 1) hessenberg_reduce_0(ctx, A, T) Q = A.copy() hessenberg_reduce_1(ctx, Q, T) for x in xrange(n): for y in xrange(x+2, n): A[y,x] = 0 return Q, A ########################################################################### def qr_step(ctx, n0, n1, A, Q, shift): """ This subroutine executes a single implicitly shifted QR step applied to an upper Hessenberg matrix A. Given A and shift as input, first an QR decomposition is calculated: Q R = A - shift * 1 . The output is then following matrix: R Q + shift * 1 parameters: n0, n1 (input) Two integers which specify the submatrix A[n0:n1,n0:n1] on which this subroutine operators. The subdiagonal elements to the left and below this submatrix must be deflated (i.e. zero). following restriction is imposed: n1>=n0+2 A (input/output) On input, A is an upper Hessenberg matrix. On output, A is replaced by "R Q + shift * 1" Q (input/output) The parameter Q is multiplied by the unitary matrix Q arising from the QR decomposition. Q can also be false, in which case the unitary matrix Q is not computated. shift (input) a complex number specifying the shift. idealy close to an eigenvalue of the bottemmost part of the submatrix A[n0:n1,n0:n1]. references: Stoer, Bulirsch - Introduction to Numerical Analysis. Kresser : Numerical Methods for General and Structured Eigenvalue Problems """ # implicitly shifted and bulge chasing is explained at p.398/399 in "Stoer, Bulirsch - Introduction to Numerical Analysis" # for bulge chasing see also "Watkins - The Matrix Eigenvalue Problem" sec.4.5,p.173 # the Givens rotation we used is determined as follows: let c,s be two complex # numbers. then we have following relation: # # v = sqrt(|c|^2 + |s|^2) # # 1/v [ c~ s~] [c] = [v] # [-s c ] [s] [0] # # the matrix on the left is our Givens rotation. n = A.rows # first step # calculate givens rotation c = A[n0 ,n0] - shift s = A[n0+1,n0] v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s))) if v == 0: v = 1 c = 1 s = 0 else: c /= v s /= v cc = ctx.conj(c) cs = ctx.conj(s) for k in xrange(n0, n): # apply givens rotation from the left x = A[n0 ,k] y = A[n0+1,k] A[n0 ,k] = cc * x + cs * y A[n0+1,k] = c * y - s * x for k in xrange(min(n1, n0+3)): # apply givens rotation from the right x = A[k,n0 ] y = A[k,n0+1] A[k,n0 ] = c * x + s * y A[k,n0+1] = cc * y - cs * x if not isinstance(Q, bool): for k in xrange(n): # eigenvectors x = Q[k,n0 ] y = Q[k,n0+1] Q[k,n0 ] = c * x + s * y Q[k,n0+1] = cc * y - cs * x # chase the bulge for j in xrange(n0, n1 - 2): # calculate givens rotation c = A[j+1,j] s = A[j+2,j] v = ctx.hypot(ctx.hypot(ctx.re(c), ctx.im(c)), ctx.hypot(ctx.re(s), ctx.im(s))) if v == 0: A[j+1,j] = 0 v = 1 c = 1 s = 0 else: A[j+1,j] = v c /= v s /= v A[j+2,j] = 0 cc = ctx.conj(c) cs = ctx.conj(s) for k in xrange(j+1, n): # apply givens rotation from the left x = A[j+1,k] y = A[j+2,k] A[j+1,k] = cc * x + cs * y A[j+2,k] = c * y - s * x for k in xrange(0, min(n1, j+4)): # apply givens rotation from the right x = A[k,j+1] y = A[k,j+2] A[k,j+1] = c * x + s * y A[k,j+2] = cc * y - cs * x if not isinstance(Q, bool): for k in xrange(0, n): # eigenvectors x = Q[k,j+1] y = Q[k,j+2] Q[k,j+1] = c * x + s * y Q[k,j+2] = cc * y - cs * x def hessenberg_qr(ctx, A, Q): """ This routine computes the Schur decomposition of an upper Hessenberg matrix A. Given A, an unitary matrix Q is determined such that Q' A Q = R and Q' Q = Q Q' = 1 where R is an upper right triangular matrix. Here ' denotes the hermitian transpose (i.e. transposition and conjugation). parameters: A (input/output) On input, A contains an upper Hessenberg matrix. On output, A is replace by the upper right triangluar matrix R. Q (input/output) The parameter Q is multiplied by the unitary matrix Q arising from the Schur decomposition. Q can also be false, in which case the unitary matrix Q is not computated. """ n = A.rows norm = 0 for x in xrange(n): for y in xrange(min(x+2, n)): norm += ctx.re(A[y,x]) ** 2 + ctx.im(A[y,x]) ** 2 norm = ctx.sqrt(norm) / n if norm == 0: return n0 = 0 n1 = n eps = ctx.eps / (100 * n) maxits = ctx.dps * 4 its = totalits = 0 while 1: # kressner p.32 algo 3 # the active submatrix is A[n0:n1,n0:n1] k = n0 while k + 1 < n1: s = abs(ctx.re(A[k,k])) + abs(ctx.im(A[k,k])) + abs(ctx.re(A[k+1,k+1])) + abs(ctx.im(A[k+1,k+1])) if s < eps * norm: s = norm if abs(A[k+1,k]) < eps * s: break k += 1 if k + 1 < n1: # deflation found at position (k+1, k) A[k+1,k] = 0 n0 = k + 1 its = 0 if n0 + 1 >= n1: # block of size at most two has converged n0 = 0 n1 = k + 1 if n1 < 2: # QR algorithm has converged return else: if (its % 30) == 10: # exceptional shift shift = A[n1-1,n1-2] elif (its % 30) == 20: # exceptional shift shift = abs(A[n1-1,n1-2]) elif (its % 30) == 29: # exceptional shift shift = norm else: # A = [ a b ] det(x-A)=x*x-x*tr(A)+det(A) # [ c d ] # # eigenvalues bad: (tr(A)+sqrt((tr(A))**2-4*det(A)))/2 # bad because of cancellation if |c| is small and |a-d| is small, too. # # eigenvalues good: (a+d+sqrt((a-d)**2+4*b*c))/2 t = A[n1-2,n1-2] + A[n1-1,n1-1] s = (A[n1-1,n1-1] - A[n1-2,n1-2]) ** 2 + 4 * A[n1-1,n1-2] * A[n1-2,n1-1] if ctx.re(s) > 0: s = ctx.sqrt(s) else: s = ctx.sqrt(-s) * 1j a = (t + s) / 2 b = (t - s) / 2 if abs(A[n1-1,n1-1] - a) > abs(A[n1-1,n1-1] - b): shift = b else: shift = a its += 1 totalits += 1 qr_step(ctx, n0, n1, A, Q, shift) if its > maxits: raise RuntimeError("qr: failed to converge after %d steps" % its) @defun def schur(ctx, A, overwrite_a = False): """ This routine computes the Schur decomposition of a square matrix A. Given A, an unitary matrix Q is determined such that Q' A Q = R and Q' Q = Q Q' = 1 where R is an upper right triangular matrix. Here ' denotes the hermitian transpose (i.e. transposition and conjugation). input: A : a real or complex square matrix overwrite_a : if true, allows modification of A which may improve performance. if false, A is not modified. output: Q : an unitary matrix R : an upper right triangular matrix return value: (Q, R) example: >>> from mpmath import mp >>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]]) >>> Q, R = mp.schur(A) >>> mp.nprint(R, 3) # doctest:+SKIP [2.0 0.417 -2.53] [0.0 4.0 -4.74] [0.0 0.0 9.0] >>> print(mp.chop(A - Q * R * Q.transpose_conj())) [0.0 0.0 0.0] [0.0 0.0 0.0] [0.0 0.0 0.0] warning: The Schur decomposition is not unique. """ n = A.rows if n == 1: return (ctx.matrix([[1]]), A) if not overwrite_a: A = A.copy() T = ctx.matrix(n, 1) hessenberg_reduce_0(ctx, A, T) Q = A.copy() hessenberg_reduce_1(ctx, Q, T) for x in xrange(n): for y in xrange(x + 2, n): A[y,x] = 0 hessenberg_qr(ctx, A, Q) return Q, A def eig_tr_r(ctx, A): """ This routine calculates the right eigenvectors of an upper right triangular matrix. input: A an upper right triangular matrix output: ER a matrix whose columns form the right eigenvectors of A return value: ER """ # this subroutine is inspired by the lapack routines ctrevc.f,clatrs.f n = A.rows ER = ctx.eye(n) eps = ctx.eps unfl = ctx.ldexp(ctx.one, -ctx.prec * 30) # since mpmath effectively has no limits on the exponent, we simply scale doubles up # original double has prec*20 smlnum = unfl * (n / eps) simin = 1 / ctx.sqrt(eps) rmax = 1 for i in xrange(1, n): s = A[i,i] smin = max(eps * abs(s), smlnum) for j in xrange(i - 1, -1, -1): r = 0 for k in xrange(j + 1, i + 1): r += A[j,k] * ER[k,i] t = A[j,j] - s if abs(t) < smin: t = smin r = -r / t ER[j,i] = r rmax = max(rmax, abs(r)) if rmax > simin: for k in xrange(j, i+1): ER[k,i] /= rmax rmax = 1 if rmax != 1: for k in xrange(0, i + 1): ER[k,i] /= rmax return ER def eig_tr_l(ctx, A): """ This routine calculates the left eigenvectors of an upper right triangular matrix. input: A an upper right triangular matrix output: EL a matrix whose rows form the left eigenvectors of A return value: EL """ n = A.rows EL = ctx.eye(n) eps = ctx.eps unfl = ctx.ldexp(ctx.one, -ctx.prec * 30) # since mpmath effectively has no limits on the exponent, we simply scale doubles up # original double has prec*20 smlnum = unfl * (n / eps) simin = 1 / ctx.sqrt(eps) rmax = 1 for i in xrange(0, n - 1): s = A[i,i] smin = max(eps * abs(s), smlnum) for j in xrange(i + 1, n): r = 0 for k in xrange(i, j): r += EL[i,k] * A[k,j] t = A[j,j] - s if abs(t) < smin: t = smin r = -r / t EL[i,j] = r rmax = max(rmax, abs(r)) if rmax > simin: for k in xrange(i, j + 1): EL[i,k] /= rmax rmax = 1 if rmax != 1: for k in xrange(i, n): EL[i,k] /= rmax return EL @defun def eig(ctx, A, left = False, right = True, overwrite_a = False): """ This routine computes the eigenvalues and optionally the left and right eigenvectors of a square matrix A. Given A, a vector E and matrices ER and EL are calculated such that A ER[:,i] = E[i] ER[:,i] EL[i,:] A = EL[i,:] E[i] E contains the eigenvalues of A. The columns of ER contain the right eigenvectors of A whereas the rows of EL contain the left eigenvectors. input: A : a real or complex square matrix of shape (n, n) left : if true, the left eigenvectors are calculated. right : if true, the right eigenvectors are calculated. overwrite_a : if true, allows modification of A which may improve performance. if false, A is not modified. output: E : a list of length n containing the eigenvalues of A. ER : a matrix whose columns contain the right eigenvectors of A. EL : a matrix whose rows contain the left eigenvectors of A. return values: E if left and right are both false. (E, ER) if right is true and left is false. (E, EL) if left is true and right is false. (E, EL, ER) if left and right are true. examples: >>> from mpmath import mp >>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]]) >>> E, ER = mp.eig(A) >>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0])) [0.0] [0.0] [0.0] >>> E, EL, ER = mp.eig(A,left = True, right = True) >>> E, EL, ER = mp.eig_sort(E, EL, ER) >>> mp.nprint(E) [2.0, 4.0, 9.0] >>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0])) [0.0] [0.0] [0.0] >>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0])) [0.0 0.0 0.0] warning: - If there are multiple eigenvalues, the eigenvectors do not necessarily span the whole vectorspace, i.e. ER and EL may have not full rank. Furthermore in that case the eigenvectors are numerical ill-conditioned. - In the general case the eigenvalues have no natural order. see also: - eigh (or eigsy, eighe) for the symmetric eigenvalue problem. - eig_sort for sorting of eigenvalues and eigenvectors """ n = A.rows if n == 1: if left and (not right): return ([A[0]], ctx.matrix([[1]])) if right and (not left): return ([A[0]], ctx.matrix([[1]])) return ([A[0]], ctx.matrix([[1]]), ctx.matrix([[1]])) if not overwrite_a: A = A.copy() T = ctx.zeros(n, 1) hessenberg_reduce_0(ctx, A, T) if left or right: Q = A.copy() hessenberg_reduce_1(ctx, Q, T) else: Q = False for x in xrange(n): for y in xrange(x + 2, n): A[y,x] = 0 hessenberg_qr(ctx, A, Q) E = [0 for i in xrange(n)] for i in xrange(n): E[i] = A[i,i] if not (left or right): return E if left: EL = eig_tr_l(ctx, A) EL = EL * Q.transpose_conj() if right: ER = eig_tr_r(ctx, A) ER = Q * ER if left and (not right): return (E, EL) if right and (not left): return (E, ER) return (E, EL, ER) @defun def eig_sort(ctx, E, EL = False, ER = False, f = "real"): """ This routine sorts the eigenvalues and eigenvectors delivered by ``eig``. parameters: E : the eigenvalues as delivered by eig EL : the left eigenvectors as delivered by eig, or false ER : the right eigenvectors as delivered by eig, or false f : either a string ("real" sort by increasing real part, "imag" sort by increasing imag part, "abs" sort by absolute value) or a function mapping complexs to the reals, i.e. ``f = lambda x: -mp.re(x) `` would sort the eigenvalues by decreasing real part. return values: E if EL and ER are both false. (E, ER) if ER is not false and left is false. (E, EL) if EL is not false and right is false. (E, EL, ER) if EL and ER are not false. example: >>> from mpmath import mp >>> A = mp.matrix([[3, -1, 2], [2, 5, -5], [-2, -3, 7]]) >>> E, EL, ER = mp.eig(A,left = True, right = True) >>> E, EL, ER = mp.eig_sort(E, EL, ER) >>> mp.nprint(E) [2.0, 4.0, 9.0] >>> E, EL, ER = mp.eig_sort(E, EL, ER,f = lambda x: -mp.re(x)) >>> mp.nprint(E) [9.0, 4.0, 2.0] >>> print(mp.chop(A * ER[:,0] - E[0] * ER[:,0])) [0.0] [0.0] [0.0] >>> print(mp.chop( EL[0,:] * A - EL[0,:] * E[0])) [0.0 0.0 0.0] """ if isinstance(f, str): if f == "real": f = ctx.re elif f == "imag": f = ctx.im elif f == "abs": f = abs else: raise RuntimeError("unknown function %s" % f) n = len(E) # Sort eigenvalues (bubble-sort) for i in xrange(n): imax = i s = f(E[i]) # s is the current maximal element for j in xrange(i + 1, n): c = f(E[j]) if c < s: s = c imax = j if imax != i: # swap eigenvalues z = E[i] E[i] = E[imax] E[imax] = z if not isinstance(EL, bool): for j in xrange(n): z = EL[i,j] EL[i,j] = EL[imax,j] EL[imax,j] = z if not isinstance(ER, bool): for j in xrange(n): z = ER[j,i] ER[j,i] = ER[j,imax] ER[j,imax] = z if isinstance(EL, bool) and isinstance(ER, bool): return E if isinstance(EL, bool) and not(isinstance(ER, bool)): return (E, ER) if isinstance(ER, bool) and not(isinstance(EL, bool)): return (E, EL) return (E, EL, ER)