''' Part of the micro-linalg project to provide a small matrix / linear algebra package for Micropython (Python3) The MIT License (MIT) Copyright (c) 2015 Jamie Lawson Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. ''' import math import umatrix def zeros(m, n, dtype=umatrix.ddtype): return umatrix.matrix([[0 for i in range(n)] for j in range(m)], dtype=dtype) def ones(m, n, dtype=umatrix.ddtype): return zeros(m, n, dtype) + 1 def eye(m, dtype=umatrix.ddtype): Z = zeros(m, m, dtype=dtype) for i in range(m): Z[i, i] = 1 return Z def det_inv(x): ''' Return (det(x) and inv(x)) Operates on a copy of x Using elementary row operations convert X to an upper matrix the product of the diagonal = det(X) Continue to convert X to the identity matrix All the operation carried out on the original identity matrix makes it the inverse of X ''' if not x.is_square: raise ValueError('Matrix must be square') else: # divide each row element by [0] to give a one in the first position # (may have to find a row to switch with if first element is 0) x = x.copy() inverse = eye(len(x), dtype=float) sign = 1 factors = [] p = 0 while p < len(x): d = x[p, p] if abs(d) < umatrix.flt_eps: # pivot == 0 need to swap a row # check if swap row also has a zero at the same position np = 1 while (p + np) < len(x) and abs(x[p + np, p]) < umatrix.flt_eps: np += 1 if (p + np) == len(x): # singular return [0, []] # swap rows z = x[p + np] x[p + np, :] = x[p] x[p, :] = z # do identity z = inverse[p + np] inverse[p + np, :] = inverse[p] inverse[p, :] = z # change sign of det sign = -sign continue factors.append(d) # change target row for n in range(p, len(x)): x[p, n] = x[p, n] / d # need to do the entire row for the inverse for n in range(len(x)): inverse[p, n] = inverse[p, n] / d # eliminate position in the following rows for i in range(p + 1, len(x)): # multiplier is that column entry t = x[i, p] for j in range(p, len(x)): x[i, j] = x[i, j] - (t * x[p, j]) for j in range(len(x)): inverse[i, j] = inverse[i, j] - (t * inverse[p, j]) p = p + 1 s = sign for i in factors: s = s * i # determinant # travel through the rows eliminating upper diagonal non-zero values for i in range(len(x) - 1): # final row should already be all zeros # except for the final position for p in range(i + 1, len(x)): # multiplier is that column entry t = x[i, p] for j in range(i + 1, len(x)): x[i, j] = x[i, j] - (t * x[p, j]) for j in range(len(x)): inverse[i, j] = inverse[i, j] - (t * inverse[p, j]) return (s, inverse) def pinv(X): ''' Calculates the pseudo inverse Adagger = (A'A)^-1.A' ''' Xt = X.transpose() d, Z = det_inv(dot(Xt, X)) return dot(Z, Xt) def dot(X, Y): ''' Dot product ''' if X.size(2) == Y.size(1): Z = [] for k in range(X.size(1)): for j in range(Y.size(2)): Z.append(sum([X[k, i] * Y[i, j] for i in range(Y.size(1))])) return umatrix.matrix(Z, cstride=1, rstride=Y.size(2)) else: raise ValueError('shapes not aligned') def cross(X, Y, axis=1): ''' Cross product axis=1 Numpy default axis=0 MATLAB, Octave, SciLab default ''' if axis == 0: X = X.T Y = Y.T if (X.n in (2, 3)) and (Y.n in (2, 3)): if X.m == Y.m: Z = [] for k in range(min(X.m, Y.m)): z = X[k, 0] * Y[k, 1] - X[k, 1] * Y[k, 0] if (X.n == 3) and (Y.n == 3): Z.append([X[k, 1] * Y[k, 2] - X[k, 2] * Y[k, 1], X[k, 2] * Y[k, 0] - X[k, 0] * Y[k, 2], z]) else: Z.append([z]) if axis == 0: return umatrix.matrix(Z).T else: return umatrix.matrix(Z) else: raise ValueError('shape mismatch') else: raise ValueError('incompatible dimensions for cross product' ' (must be 2 or 3)') def eps(x = 0): # ref. numpy.spacing(), Octave/MATLAB eps() function if x: return 2**(math.floor(math.log(abs(x))/math.log(2)))*umatrix.flt_eps else: return umatrix.flt_eps