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