260 lines
6.3 KiB
Python
260 lines
6.3 KiB
Python
import sys
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import numpy as np
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from functools import wraps
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import itertools
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import logging
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log = logging.getLogger('interp')
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def get_phis(X, R):
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"""
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The get_phis function is used to get barycentric coordonites for a point on
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a triangle or tetrahedron:
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in 2D:
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X -- the destination point (2D)
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X = [0,0]
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r -- the three points that make up the containing triangular simplex (2D)
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r = [[-1, -1], [0, 2], [1, -1]]
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this will return [0.333, 0.333, 0.333]
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in 3D:
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X -- the destination point (3D)
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X = [0,0,0]
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R -- the four points that make up the containing simplex, tetrahedron (3D)
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R = [
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[0.0, 0.0, 1.0],
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[0.94280904333606508, 0.0, -0.3333333283722672],
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[-0.47140452166803232, 0.81649658244673617, -0.3333333283722672],
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[-0.47140452166803298, -0.81649658244673584, -0.3333333283722672],
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]
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this will return [0.25, 0.25, 0.25, 0.25]
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"""
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# baker: eq 7
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# TODO: perhaps also test len(R[0]) .. ?
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if len(X) == 2:
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log.debug("running 2D")
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A = np.array([
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[ 1, 1, 1],
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[R[0][0], R[1][0], R[2][0]],
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[R[0][1], R[1][1], R[2][1]],
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])
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b = np.array([ 1,
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X[0],
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X[1]
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])
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elif len(X) == 3:
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log.debug("running 3D")
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A = np.array([
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[ 1, 1, 1, 1 ],
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[R[0][0], R[1][0], R[2][0], R[3][0]],
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[R[0][1], R[1][1], R[2][1], R[3][1]],
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[R[0][2], R[1][2], R[2][2], R[3][2]],
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])
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b = np.array([ 1,
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X[0],
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X[1],
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X[2]
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])
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else:
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raise Exception("inapropriate demension on X")
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try:
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phi = np.linalg.solve(A,b)
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except np.linalg.LinAlgError as e:
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msg = "calculation of phis yielded a linearly dependant system (%s)" % e
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log.error(msg)
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# raise Exception(msg)
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phi = np.dot(np.linalg.pinv(A), b)
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log.debug("phi: %s", phi)
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return phi
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def qlinear(X, R):
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"""
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this calculates the linear portion of q from R to X
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also, this is baker eq 3
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X = destination point
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R = a inter.grid object; must have R.points and R.q
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"""
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phis = get_phis(X, R.verts)
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qlin = np.sum([q_i * phi_i for q_i, phi_i in zip(R.q, phis)])
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log.debug("phis: %s", phis)
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log.debug("qlin: %s", qlin)
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return phis, qlin
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def get_error(phi, R, S, order = 2):
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B = [] # baker eq 9
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w = [] # baker eq 11
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p = pattern(len(phi), order)
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log.info("pattern: %s" % p)
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for (s,q) in zip(S.verts, S.q):
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cur_phi, cur_qlin = qlinear(s, R)
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l = []
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for i in p:
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cur_sum = cur_phi[i[0]]
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for j in i[1:]:
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cur_sum *= cur_phi[j]
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l.append(cur_sum)
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B.append(l)
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w.append(q - cur_qlin)
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log.info("B: %s" % B)
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log.info("w: %s" % w)
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B = np.array(B)
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w = np.array(w)
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A = np.dot(B.T, B)
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b = np.dot(B.T, w)
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# baker solve eq 10
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try:
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abc = np.linalg.solve(A,b)
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except np.linalg.LinAlgError as e:
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log.error("linear calculation went bad, resorting to np.linalg.pinv: %s" % e)
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abc = np.dot(np.linalg.pinv(A), b)
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error_term = 0.0
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for (a, i) in zip(abc, p):
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cur_sum = a
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for j in i:
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cur_sum *= phi[j]
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error_term += cur_sum
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log.debug("error_term: %s" % error_term)
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return error_term, abc
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def run_baker(X, R, S, order=2):
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"""
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This is the main function to call to get an interpolation to X from the
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input meshes
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X -- the destination point (2D)
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X = [0,0]
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R = Simplex
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S = extra points
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"""
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log.debug("order = %d" % order)
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log.debug("extra points = %d" % len(S.verts))
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answer = {
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'qlin': None,
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'error': None,
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'final': None,
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}
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# calculate values only for the simplex triangle
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phi, qlin = qlinear(X, R)
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if order == 1:
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answer['qlin'] = qlin
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return answer
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elif order in xrange(2,11):
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error_term, abc = get_error(phi, R, S, order)
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else:
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raise Exception('unsupported order "%d" for baker method' % order)
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q_final = qlin + error_term
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answer['qlin' ] = qlin
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answer['error'] = error_term
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answer['final'] = q_final
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answer['abc' ] = abc
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log.debug(answer)
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return answer
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def memoize(f):
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"""
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for more information on what I'm doing here,
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please read:
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http://en.wikipedia.org/wiki/Memoize
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"""
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cache = {}
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@wraps(f)
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def memf(*x, **kargs):
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if x not in cache:
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cache[x] = f(*x, **kargs)
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return cache[x]
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return memf
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def combinations_with_replacement(iterable, r):
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"""
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What I really need for the pattern function only
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exists in python 2.7 and greater. The docs suggest
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the implementation in this function as a
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replacement.
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see: http://docs.python.org/library/itertools.html#itertools.combinations_with_replacement
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"""
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pool = tuple(iterable)
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n = len(pool)
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for indices in itertools.product(range(n), repeat=r):
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if sorted(indices) == list(indices):
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yield tuple(pool[i] for i in indices)
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@memoize
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def pattern(simplex_size, nu):
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"""
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my useful docstring
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"""
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log.debug("pattern: simplex: %d, order: %d" % (simplex_size, nu))
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return [i for i in combinations_with_replacement(xrange(simplex_size), nu) if len(set(i)) != 1]
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if __name__ == '__main__':
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print len(pattern(3, 2)), pattern(3, 2)
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print len(pattern(4, 2)), pattern(4, 2)
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print len(pattern(3, 3)), pattern(3, 3)
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print len(pattern(4, 3)), pattern(4, 3)
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print len(pattern(3, 4)), pattern(3, 4)
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print len(pattern(4, 4)), pattern(4, 4)
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print len(pattern(3, 2)), pattern(3, 2)
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print len(pattern(4, 2)), pattern(4, 2)
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print len(pattern(3, 3)), pattern(3, 3)
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print len(pattern(4, 3)), pattern(4, 3)
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print len(pattern(3, 4)), pattern(3, 4)
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print len(pattern(4, 4)), pattern(4, 4)
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print len(pattern(3, 2)), pattern(3, 2)
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print len(pattern(4, 2)), pattern(4, 2)
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print len(pattern(3, 3)), pattern(3, 3)
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print len(pattern(4, 3)), pattern(4, 3)
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print len(pattern(3, 4)), pattern(3, 4)
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print len(pattern(4, 4)), pattern(4, 4)
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print len(pattern(3, 2)), pattern(3, 2)
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print len(pattern(4, 2)), pattern(4, 2)
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print len(pattern(3, 3)), pattern(3, 3)
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print len(pattern(4, 3)), pattern(4, 3)
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print len(pattern(3, 4)), pattern(3, 4)
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print len(pattern(4, 4)), pattern(4, 4)
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