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from ..libmp.backend import xrange |
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class MatrixCalculusMethods(object): |
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def _exp_pade(ctx, a): |
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""" |
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Exponential of a matrix using Pade approximants. |
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See G. H. Golub, C. F. van Loan 'Matrix Computations', |
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third Ed., page 572 |
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TODO: |
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- find a good estimate for q |
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- reduce the number of matrix multiplications to improve |
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performance |
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""" |
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def eps_pade(p): |
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return ctx.mpf(2)**(3-2*p) * \ |
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ctx.factorial(p)**2/(ctx.factorial(2*p)**2 * (2*p + 1)) |
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q = 4 |
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extraq = 8 |
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while 1: |
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if eps_pade(q) < ctx.eps: |
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break |
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q += 1 |
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q += extraq |
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j = int(max(1, ctx.mag(ctx.mnorm(a,'inf')))) |
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extra = q |
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prec = ctx.prec |
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ctx.dps += extra + 3 |
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try: |
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a = a/2**j |
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na = a.rows |
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den = ctx.eye(na) |
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num = ctx.eye(na) |
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x = ctx.eye(na) |
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c = ctx.mpf(1) |
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for k in range(1, q+1): |
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c *= ctx.mpf(q - k + 1)/((2*q - k + 1) * k) |
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x = a*x |
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cx = c*x |
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num += cx |
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den += (-1)**k * cx |
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f = ctx.lu_solve_mat(den, num) |
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for k in range(j): |
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f = f*f |
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finally: |
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ctx.prec = prec |
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return f*1 |
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def expm(ctx, A, method='taylor'): |
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r""" |
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Computes the matrix exponential of a square matrix `A`, which is defined |
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by the power series |
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.. math :: |
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\exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots |
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With method='taylor', the matrix exponential is computed |
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using the Taylor series. With method='pade', Pade approximants |
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are used instead. |
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**Examples** |
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Basic examples:: |
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>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> expm(zeros(3)) |
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[1.0 0.0 0.0] |
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[0.0 1.0 0.0] |
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[0.0 0.0 1.0] |
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>>> expm(eye(3)) |
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[2.71828182845905 0.0 0.0] |
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[ 0.0 2.71828182845905 0.0] |
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[ 0.0 0.0 2.71828182845905] |
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>>> expm([[1,1,0],[1,0,1],[0,1,0]]) |
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[ 3.86814500615414 2.26812870852145 0.841130841230196] |
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[ 2.26812870852145 2.44114713886289 1.42699786729125] |
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[0.841130841230196 1.42699786729125 1.6000162976327] |
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>>> expm([[1,1,0],[1,0,1],[0,1,0]], method='pade') |
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[ 3.86814500615414 2.26812870852145 0.841130841230196] |
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[ 2.26812870852145 2.44114713886289 1.42699786729125] |
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[0.841130841230196 1.42699786729125 1.6000162976327] |
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>>> expm([[1+j, 0], [1+j,1]]) |
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[(1.46869393991589 + 2.28735528717884j) 0.0] |
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[ (1.03776739863568 + 3.536943175722j) (2.71828182845905 + 0.0j)] |
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Matrices with large entries are allowed:: |
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>>> expm(matrix([[1,2],[2,3]])**25) |
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[5.65024064048415e+2050488462815550 9.14228140091932e+2050488462815550] |
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[9.14228140091932e+2050488462815550 1.47925220414035e+2050488462815551] |
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The identity `\exp(A+B) = \exp(A) \exp(B)` does not hold for |
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noncommuting matrices:: |
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>>> A = hilbert(3) |
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>>> B = A + eye(3) |
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>>> chop(mnorm(A*B - B*A)) |
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0.0 |
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>>> chop(mnorm(expm(A+B) - expm(A)*expm(B))) |
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0.0 |
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>>> B = A + ones(3) |
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>>> mnorm(A*B - B*A) |
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1.8 |
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>>> mnorm(expm(A+B) - expm(A)*expm(B)) |
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42.0927851137247 |
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""" |
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if method == 'pade': |
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prec = ctx.prec |
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try: |
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A = ctx.matrix(A) |
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ctx.prec += 2*A.rows |
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res = ctx._exp_pade(A) |
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finally: |
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ctx.prec = prec |
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return res |
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A = ctx.matrix(A) |
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prec = ctx.prec |
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j = int(max(1, ctx.mag(ctx.mnorm(A,'inf')))) |
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j += int(0.5*prec**0.5) |
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try: |
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ctx.prec += 10 + 2*j |
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tol = +ctx.eps |
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A = A/2**j |
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T = A |
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Y = A**0 + A |
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k = 2 |
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while 1: |
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T *= A * (1/ctx.mpf(k)) |
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if ctx.mnorm(T, 'inf') < tol: |
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break |
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Y += T |
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k += 1 |
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for k in xrange(j): |
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Y = Y*Y |
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finally: |
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ctx.prec = prec |
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Y *= 1 |
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return Y |
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def cosm(ctx, A): |
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r""" |
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Gives the cosine of a square matrix `A`, defined in analogy |
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with the matrix exponential. |
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Examples:: |
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>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> X = eye(3) |
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>>> cosm(X) |
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[0.54030230586814 0.0 0.0] |
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[ 0.0 0.54030230586814 0.0] |
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[ 0.0 0.0 0.54030230586814] |
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>>> X = hilbert(3) |
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>>> cosm(X) |
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[ 0.424403834569555 -0.316643413047167 -0.221474945949293] |
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[-0.316643413047167 0.820646708837824 -0.127183694770039] |
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[-0.221474945949293 -0.127183694770039 0.909236687217541] |
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>>> X = matrix([[1+j,-2],[0,-j]]) |
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>>> cosm(X) |
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[(0.833730025131149 - 0.988897705762865j) (1.07485840848393 - 0.17192140544213j)] |
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[ 0.0 (1.54308063481524 + 0.0j)] |
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""" |
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B = 0.5 * (ctx.expm(A*ctx.j) + ctx.expm(A*(-ctx.j))) |
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if not sum(A.apply(ctx.im).apply(abs)): |
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B = B.apply(ctx.re) |
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return B |
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def sinm(ctx, A): |
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r""" |
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Gives the sine of a square matrix `A`, defined in analogy |
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with the matrix exponential. |
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Examples:: |
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>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> X = eye(3) |
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>>> sinm(X) |
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[0.841470984807897 0.0 0.0] |
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[ 0.0 0.841470984807897 0.0] |
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[ 0.0 0.0 0.841470984807897] |
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>>> X = hilbert(3) |
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>>> sinm(X) |
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[0.711608512150994 0.339783913247439 0.220742837314741] |
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[0.339783913247439 0.244113865695532 0.187231271174372] |
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[0.220742837314741 0.187231271174372 0.155816730769635] |
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>>> X = matrix([[1+j,-2],[0,-j]]) |
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>>> sinm(X) |
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[(1.29845758141598 + 0.634963914784736j) (-1.96751511930922 + 0.314700021761367j)] |
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[ 0.0 (0.0 - 1.1752011936438j)] |
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""" |
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B = (-0.5j) * (ctx.expm(A*ctx.j) - ctx.expm(A*(-ctx.j))) |
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if not sum(A.apply(ctx.im).apply(abs)): |
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B = B.apply(ctx.re) |
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return B |
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def _sqrtm_rot(ctx, A, _may_rotate): |
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u = ctx.j**0.3 |
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return ctx.sqrtm(u*A, _may_rotate) / ctx.sqrt(u) |
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def sqrtm(ctx, A, _may_rotate=2): |
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r""" |
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Computes a square root of the square matrix `A`, i.e. returns |
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a matrix `B = A^{1/2}` such that `B^2 = A`. The square root |
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of a matrix, if it exists, is not unique. |
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|
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**Examples** |
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Square roots of some simple matrices:: |
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>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> sqrtm([[1,0], [0,1]]) |
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[1.0 0.0] |
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[0.0 1.0] |
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>>> sqrtm([[0,0], [0,0]]) |
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[0.0 0.0] |
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[0.0 0.0] |
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>>> sqrtm([[2,0],[0,1]]) |
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[1.4142135623731 0.0] |
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[ 0.0 1.0] |
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>>> sqrtm([[1,1],[1,0]]) |
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[ (0.920442065259926 - 0.21728689675164j) (0.568864481005783 + 0.351577584254143j)] |
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[(0.568864481005783 + 0.351577584254143j) (0.351577584254143 - 0.568864481005783j)] |
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>>> sqrtm([[1,0],[0,1]]) |
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[1.0 0.0] |
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[0.0 1.0] |
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>>> sqrtm([[-1,0],[0,1]]) |
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[(0.0 - 1.0j) 0.0] |
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[ 0.0 (1.0 + 0.0j)] |
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>>> sqrtm([[j,0],[0,j]]) |
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[(0.707106781186547 + 0.707106781186547j) 0.0] |
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[ 0.0 (0.707106781186547 + 0.707106781186547j)] |
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A square root of a rotation matrix, giving the corresponding |
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half-angle rotation matrix:: |
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>>> t1 = 0.75 |
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>>> t2 = t1 * 0.5 |
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>>> A1 = matrix([[cos(t1), -sin(t1)], [sin(t1), cos(t1)]]) |
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>>> A2 = matrix([[cos(t2), -sin(t2)], [sin(t2), cos(t2)]]) |
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>>> sqrtm(A1) |
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[0.930507621912314 -0.366272529086048] |
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[0.366272529086048 0.930507621912314] |
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>>> A2 |
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[0.930507621912314 -0.366272529086048] |
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[0.366272529086048 0.930507621912314] |
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The identity `(A^2)^{1/2} = A` does not necessarily hold:: |
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>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]]) |
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>>> sqrtm(A**2) |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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>>> sqrtm(A)**2 |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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>>> A = matrix([[-4,1,4],[7,-8,9],[10,2,11]]) |
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>>> sqrtm(A**2) |
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[ 7.43715112194995 -0.324127569985474 1.8481718827526] |
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[-0.251549715716942 9.32699765900402 2.48221180985147] |
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[ 4.11609388833616 0.775751877098258 13.017955697342] |
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>>> chop(sqrtm(A)**2) |
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[-4.0 1.0 4.0] |
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[ 7.0 -8.0 9.0] |
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[10.0 2.0 11.0] |
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For some matrices, a square root does not exist:: |
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>>> sqrtm([[0,1], [0,0]]) |
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Traceback (most recent call last): |
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... |
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ZeroDivisionError: matrix is numerically singular |
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|
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Two examples from the documentation for Matlab's ``sqrtm``:: |
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>>> mp.dps = 15; mp.pretty = True |
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>>> sqrtm([[7,10],[15,22]]) |
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[1.56669890360128 1.74077655955698] |
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[2.61116483933547 4.17786374293675] |
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>>> |
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>>> X = matrix(\ |
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... [[5,-4,1,0,0], |
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... [-4,6,-4,1,0], |
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... [1,-4,6,-4,1], |
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... [0,1,-4,6,-4], |
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... [0,0,1,-4,5]]) |
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>>> Y = matrix(\ |
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... [[2,-1,-0,-0,-0], |
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... [-1,2,-1,0,-0], |
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... [0,-1,2,-1,0], |
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... [-0,0,-1,2,-1], |
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... [-0,-0,-0,-1,2]]) |
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>>> mnorm(sqrtm(X) - Y) |
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4.53155328326114e-19 |
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""" |
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A = ctx.matrix(A) |
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if A*0 == A: |
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return A |
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prec = ctx.prec |
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if _may_rotate: |
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d = ctx.det(A) |
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if abs(ctx.im(d)) < 16*ctx.eps and ctx.re(d) < 0: |
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return ctx._sqrtm_rot(A, _may_rotate-1) |
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try: |
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ctx.prec += 10 |
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tol = ctx.eps * 128 |
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Y = A |
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Z = I = A**0 |
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k = 0 |
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while 1: |
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Yprev = Y |
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try: |
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Y, Z = 0.5*(Y+ctx.inverse(Z)), 0.5*(Z+ctx.inverse(Y)) |
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except ZeroDivisionError: |
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if _may_rotate: |
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Y = ctx._sqrtm_rot(A, _may_rotate-1) |
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break |
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else: |
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raise |
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mag1 = ctx.mnorm(Y-Yprev, 'inf') |
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mag2 = ctx.mnorm(Y, 'inf') |
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if mag1 <= mag2*tol: |
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break |
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if _may_rotate and k > 6 and not mag1 < mag2 * 0.001: |
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return ctx._sqrtm_rot(A, _may_rotate-1) |
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k += 1 |
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if k > ctx.prec: |
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raise ctx.NoConvergence |
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finally: |
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ctx.prec = prec |
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Y *= 1 |
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return Y |
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|
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def logm(ctx, A): |
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r""" |
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Computes a logarithm of the square matrix `A`, i.e. returns |
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a matrix `B = \log(A)` such that `\exp(B) = A`. The logarithm |
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of a matrix, if it exists, is not unique. |
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|
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**Examples** |
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Logarithms of some simple matrices:: |
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|
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>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> X = eye(3) |
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>>> logm(X) |
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[0.0 0.0 0.0] |
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[0.0 0.0 0.0] |
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[0.0 0.0 0.0] |
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>>> logm(2*X) |
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[0.693147180559945 0.0 0.0] |
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[ 0.0 0.693147180559945 0.0] |
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[ 0.0 0.0 0.693147180559945] |
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>>> logm(expm(X)) |
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[1.0 0.0 0.0] |
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[0.0 1.0 0.0] |
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[0.0 0.0 1.0] |
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|
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A logarithm of a complex matrix:: |
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>>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]]) |
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>>> B = logm(X) |
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>>> nprint(B) |
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[ (0.808757 + 0.107759j) (2.20752 + 0.202762j) (1.07376 - 0.773874j)] |
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[ (0.905709 - 0.107795j) (0.0287395 - 0.824993j) (0.111619 + 0.514272j)] |
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[(-0.930151 + 0.399512j) (-2.06266 - 0.674397j) (0.791552 + 0.519839j)] |
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>>> chop(expm(B)) |
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[(2.0 + 1.0j) 1.0 3.0] |
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[(1.0 - 1.0j) (1.0 - 2.0j) 1.0] |
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[ -4.0 -5.0 (0.0 + 1.0j)] |
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|
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A matrix `X` close to the identity matrix, for which |
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`\log(\exp(X)) = \exp(\log(X)) = X` holds:: |
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|
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>>> X = eye(3) + hilbert(3)/4 |
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>>> X |
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[ 1.25 0.125 0.0833333333333333] |
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[ 0.125 1.08333333333333 0.0625] |
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[0.0833333333333333 0.0625 1.05] |
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>>> logm(expm(X)) |
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[ 1.25 0.125 0.0833333333333333] |
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[ 0.125 1.08333333333333 0.0625] |
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[0.0833333333333333 0.0625 1.05] |
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>>> expm(logm(X)) |
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[ 1.25 0.125 0.0833333333333333] |
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[ 0.125 1.08333333333333 0.0625] |
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[0.0833333333333333 0.0625 1.05] |
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|
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A logarithm of a rotation matrix, giving back the angle of |
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the rotation:: |
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|
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>>> t = 3.7 |
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>>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]]) |
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>>> chop(logm(A)) |
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[ 0.0 -2.58318530717959] |
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[2.58318530717959 0.0] |
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>>> (2*pi-t) |
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2.58318530717959 |
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|
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For some matrices, a logarithm does not exist:: |
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|
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>>> logm([[1,0], [0,0]]) |
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Traceback (most recent call last): |
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... |
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ZeroDivisionError: matrix is numerically singular |
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|
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Logarithm of a matrix with large entries:: |
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|
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>>> logm(hilbert(3) * 10**20).apply(re) |
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[ 45.5597513593433 1.27721006042799 0.317662687717978] |
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[ 1.27721006042799 42.5222778973542 2.24003708791604] |
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[0.317662687717978 2.24003708791604 42.395212822267] |
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|
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""" |
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A = ctx.matrix(A) |
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prec = ctx.prec |
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try: |
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ctx.prec += 10 |
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tol = ctx.eps * 128 |
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I = A**0 |
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B = A |
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n = 0 |
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while 1: |
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B = ctx.sqrtm(B) |
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n += 1 |
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if ctx.mnorm(B-I, 'inf') < 0.125: |
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break |
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T = X = B-I |
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L = X*0 |
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k = 1 |
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while 1: |
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if k & 1: |
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L += T / k |
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else: |
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L -= T / k |
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T *= X |
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if ctx.mnorm(T, 'inf') < tol: |
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break |
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k += 1 |
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if k > ctx.prec: |
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raise ctx.NoConvergence |
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finally: |
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ctx.prec = prec |
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L *= 2**n |
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return L |
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|
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def powm(ctx, A, r): |
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r""" |
|
Computes `A^r = \exp(A \log r)` for a matrix `A` and complex |
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number `r`. |
|
|
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**Examples** |
|
|
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Powers and inverse powers of a matrix:: |
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|
|
>>> from mpmath import * |
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>>> mp.dps = 15; mp.pretty = True |
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>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]]) |
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>>> powm(A, 2) |
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[ 63.0 20.0 69.0] |
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[174.0 89.0 199.0] |
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[164.0 48.0 179.0] |
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>>> chop(powm(powm(A, 4), 1/4.)) |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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>>> powm(extraprec(20)(powm)(A, -4), -1/4.) |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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>>> chop(powm(powm(A, 1+0.5j), 1/(1+0.5j))) |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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>>> powm(extraprec(5)(powm)(A, -1.5), -1/(1.5)) |
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[ 4.0 1.0 4.0] |
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[ 7.0 8.0 9.0] |
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[10.0 2.0 11.0] |
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|
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A Fibonacci-generating matrix:: |
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|
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>>> powm([[1,1],[1,0]], 10) |
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[89.0 55.0] |
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[55.0 34.0] |
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>>> fib(10) |
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55.0 |
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>>> powm([[1,1],[1,0]], 6.5) |
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[(16.5166626964253 - 0.0121089837381789j) (10.2078589271083 + 0.0195927472575932j)] |
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[(10.2078589271083 + 0.0195927472575932j) (6.30880376931698 - 0.0317017309957721j)] |
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>>> (phi**6.5 - (1-phi)**6.5)/sqrt(5) |
|
(10.2078589271083 - 0.0195927472575932j) |
|
>>> powm([[1,1],[1,0]], 6.2) |
|
[ (14.3076953002666 - 0.008222855781077j) (8.81733464837593 + 0.0133048601383712j)] |
|
[(8.81733464837593 + 0.0133048601383712j) (5.49036065189071 - 0.0215277159194482j)] |
|
>>> (phi**6.2 - (1-phi)**6.2)/sqrt(5) |
|
(8.81733464837593 - 0.0133048601383712j) |
|
|
|
""" |
|
A = ctx.matrix(A) |
|
r = ctx.convert(r) |
|
prec = ctx.prec |
|
try: |
|
ctx.prec += 10 |
|
if ctx.isint(r): |
|
v = A ** int(r) |
|
elif ctx.isint(r*2): |
|
y = int(r*2) |
|
v = ctx.sqrtm(A) ** y |
|
else: |
|
v = ctx.expm(r*ctx.logm(A)) |
|
finally: |
|
ctx.prec = prec |
|
v *= 1 |
|
return v |
|
|
