[5897] | 1 | #!/usr/bin/env python |
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| 2 | """Auxiliary numerical tools |
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| 3 | |
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| 4 | """ |
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| 5 | |
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| 6 | from math import acos, pi, sqrt |
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| 7 | from warnings import warn |
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| 8 | |
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[6158] | 9 | import Numeric as num |
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[5897] | 10 | |
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[6158] | 11 | NAN = (num.array([1])/0.)[0] |
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[5897] | 12 | # if we use a package that has NAN, this should be updated to use NAN. |
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| 13 | |
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| 14 | # Static variable used by get_machine_precision |
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| 15 | machine_precision = None |
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| 16 | |
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| 17 | |
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| 18 | def safe_acos(x): |
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| 19 | """Safely compute acos |
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| 20 | |
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| 21 | Protect against cases where input argument x is outside the allowed |
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| 22 | interval [-1.0, 1.0] by no more than machine precision |
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| 23 | """ |
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| 24 | |
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| 25 | error_msg = 'Input to acos is outside allowed domain [-1.0, 1.0].'+\ |
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| 26 | 'I got %.12f' %x |
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| 27 | warning_msg = 'Changing argument to acos from %.18f to %.1f' %(x, sign(x)) |
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| 28 | |
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| 29 | eps = get_machine_precision() # Machine precision |
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| 30 | if x < -1.0: |
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| 31 | if x < -1.0 - eps: |
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| 32 | raise ValueError, errmsg |
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| 33 | else: |
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| 34 | warn(warning_msg) |
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| 35 | x = -1.0 |
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| 36 | |
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| 37 | if x > 1.0: |
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| 38 | if x > 1.0 + eps: |
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| 39 | raise ValueError, errmsg |
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| 40 | else: |
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| 41 | print 'NOTE: changing argument to acos from %.18f to 1.0' %x |
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| 42 | x = 1.0 |
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| 43 | |
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| 44 | return acos(x) |
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| 45 | |
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| 46 | |
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| 47 | def sign(x): |
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| 48 | if x > 0: return 1 |
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| 49 | if x < 0: return -1 |
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| 50 | if x == 0: return 0 |
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| 51 | |
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| 52 | |
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| 53 | def is_scalar(x): |
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| 54 | """True if x is a scalar (constant numeric value) |
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| 55 | """ |
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| 56 | |
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| 57 | from types import IntType, FloatType |
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| 58 | if type(x) in [IntType, FloatType]: |
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| 59 | return True |
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| 60 | else: |
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| 61 | return False |
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| 62 | |
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| 63 | def angle(v1, v2=None): |
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| 64 | """Compute angle between 2D vectors v1 and v2. |
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| 65 | |
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| 66 | If v2 is not specified it will default |
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| 67 | to e1 (the unit vector in the x-direction) |
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| 68 | |
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| 69 | The angle is measured as a number in [0, 2pi] from v2 to v1. |
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| 70 | """ |
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| 71 | |
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| 72 | # Prepare two Numeric vectors |
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| 73 | if v2 is None: |
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| 74 | v2 = [1.0, 0.0] # Unit vector along the x-axis |
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| 75 | |
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[6158] | 76 | v1 = ensure_numeric(v1, num.Float) |
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| 77 | v2 = ensure_numeric(v2, num.Float) |
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[5897] | 78 | |
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| 79 | # Normalise |
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[6158] | 80 | v1 = v1/num.sqrt(num.sum(v1**2)) |
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| 81 | v2 = v2/num.sqrt(num.sum(v2**2)) |
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[5897] | 82 | |
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| 83 | # Compute angle |
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[6158] | 84 | p = num.innerproduct(v1, v2) |
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| 85 | c = num.innerproduct(v1, normal_vector(v2)) # Projection onto normal |
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[5897] | 86 | # (negative cross product) |
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| 87 | |
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| 88 | theta = safe_acos(p) |
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| 89 | |
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| 90 | |
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| 91 | # Correct if v1 is in quadrant 3 or 4 with respect to v2 (as the x-axis) |
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| 92 | # If v2 was the unit vector [1,0] this would correspond to the test |
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| 93 | # if v1[1] < 0: theta = 2*pi-theta |
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| 94 | # In general we use the sign of the projection onto the normal. |
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| 95 | if c < 0: |
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| 96 | #Quadrant 3 or 4 |
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| 97 | theta = 2*pi-theta |
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| 98 | |
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| 99 | return theta |
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| 100 | |
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| 101 | |
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| 102 | def anglediff(v0, v1): |
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| 103 | """Compute difference between angle of vector v0 (x0, y0) and v1 (x1, y1). |
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| 104 | This is used for determining the ordering of vertices, |
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| 105 | e.g. for checking if they are counter clockwise. |
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| 106 | |
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| 107 | Always return a positive value |
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| 108 | """ |
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| 109 | |
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| 110 | from math import pi |
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| 111 | |
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| 112 | a0 = angle(v0) |
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| 113 | a1 = angle(v1) |
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| 114 | |
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| 115 | #Ensure that difference will be positive |
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| 116 | if a0 < a1: |
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| 117 | a0 += 2*pi |
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| 118 | |
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| 119 | return a0-a1 |
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| 120 | |
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| 121 | def normal_vector(v): |
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| 122 | """Normal vector to v. |
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| 123 | |
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| 124 | Returns vector 90 degrees counter clockwise to and of same length as v |
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| 125 | """ |
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| 126 | |
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[6158] | 127 | return num.array([-v[1], v[0]], num.Float) |
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[5897] | 128 | |
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| 129 | |
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| 130 | #def crossproduct_length(v1, v2): |
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| 131 | # return v1[0]*v2[1]-v2[0]*v1[1] |
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| 132 | |
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| 133 | |
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| 134 | def mean(x): |
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| 135 | """Mean value of a vector |
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| 136 | """ |
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[6158] | 137 | return(float(num.sum(x))/len(x)) |
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[5897] | 138 | |
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| 139 | |
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| 140 | def cov(x, y=None): |
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| 141 | """Covariance of vectors x and y. |
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| 142 | |
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| 143 | If y is None: return cov(x, x) |
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| 144 | """ |
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| 145 | |
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| 146 | if y is None: |
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| 147 | y = x |
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| 148 | |
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| 149 | x = ensure_numeric(x) |
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| 150 | y = ensure_numeric(y) |
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| 151 | msg = 'Lengths must be equal: len(x) == %d, len(y) == %d' %(len(x), len(y)) |
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| 152 | assert(len(x)==len(y)), msg |
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| 153 | |
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| 154 | N = len(x) |
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| 155 | cx = x - mean(x) |
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| 156 | cy = y - mean(y) |
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| 157 | |
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[6158] | 158 | p = num.innerproduct(cx,cy)/N |
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[5897] | 159 | return(p) |
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| 160 | |
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| 161 | |
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| 162 | def err(x, y=0, n=2, relative=True): |
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| 163 | """Relative error of ||x-y|| to ||y|| |
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| 164 | n = 2: Two norm |
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| 165 | n = None: Max norm |
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| 166 | |
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| 167 | If denominator evaluates to zero or |
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| 168 | if y is omitted or |
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| 169 | if keyword relative is False, |
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| 170 | absolute error is returned |
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| 171 | |
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| 172 | If there is x and y, n=2 and relative=False, this will calc; |
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| 173 | sqrt(sum_over_x&y((xi - yi)^2)) |
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| 174 | |
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| 175 | Given this value (err), to calc the root mean square deviation, do |
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| 176 | err/sqrt(n) |
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| 177 | where n is the number of elements,(len(x)) |
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| 178 | """ |
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| 179 | |
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| 180 | x = ensure_numeric(x) |
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| 181 | if y: |
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| 182 | y = ensure_numeric(y) |
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| 183 | |
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| 184 | if n == 2: |
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| 185 | err = norm(x-y) |
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| 186 | if relative is True: |
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| 187 | try: |
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| 188 | err = err/norm(y) |
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| 189 | except: |
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| 190 | pass |
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| 191 | |
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| 192 | else: |
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| 193 | err = max(abs(x-y)) |
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| 194 | if relative is True: |
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| 195 | try: |
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| 196 | err = err/max(abs(y)) |
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| 197 | except: |
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| 198 | pass |
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| 199 | |
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| 200 | return err |
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| 201 | |
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| 202 | |
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| 203 | def norm(x): |
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| 204 | """2-norm of x |
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| 205 | """ |
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| 206 | |
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[6158] | 207 | y = num.ravel(x) |
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| 208 | p = num.sqrt(num.innerproduct(y,y)) |
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[5897] | 209 | return p |
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| 210 | |
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| 211 | |
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| 212 | def corr(x, y=None): |
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| 213 | """Correlation of x and y |
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| 214 | If y is None return autocorrelation of x |
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| 215 | """ |
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| 216 | |
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| 217 | from math import sqrt |
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| 218 | if y is None: |
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| 219 | y = x |
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| 220 | |
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| 221 | varx = cov(x) |
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| 222 | vary = cov(y) |
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| 223 | |
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| 224 | if varx == 0 or vary == 0: |
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| 225 | C = 0 |
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| 226 | else: |
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| 227 | C = cov(x,y)/sqrt(varx * vary) |
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| 228 | |
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| 229 | return(C) |
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| 230 | |
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| 231 | |
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| 232 | |
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[6436] | 233 | def ensure_numeric(A, typecode=None): |
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| 234 | """Ensure that sequence is a numeric array. |
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| 235 | |
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[5897] | 236 | Inputs: |
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| 237 | A: Sequence. If A is already a Numeric array it will be returned |
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| 238 | unaltered |
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| 239 | If not, an attempt is made to convert it to a Numeric |
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| 240 | array |
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| 241 | A: Scalar. Return 0-dimensional array of length 1, containing that value |
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| 242 | A: String. Array of ASCII values |
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| 243 | typecode: Numeric type. If specified, use this in the conversion. |
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| 244 | If not, let Numeric decide |
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| 245 | |
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| 246 | This function is necessary as array(A) can cause memory overflow. |
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| 247 | """ |
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| 248 | |
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| 249 | if typecode is None: |
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[6158] | 250 | if type(A) == num.ArrayType: |
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[5897] | 251 | return A |
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| 252 | else: |
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[6158] | 253 | return num.array(A) |
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[5897] | 254 | else: |
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[6158] | 255 | if type(A) == num.ArrayType: |
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[6438] | 256 | if A.typecode() == typecode: |
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[6174] | 257 | return A |
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[5897] | 258 | else: |
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[6436] | 259 | return num.array(A, typecode) |
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| 260 | else: |
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| 261 | return num.array(A, typecode) |
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[5897] | 262 | |
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| 263 | |
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| 264 | |
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| 265 | |
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| 266 | def histogram(a, bins, relative=False): |
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| 267 | """Standard histogram straight from the Numeric manual |
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| 268 | |
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| 269 | If relative is True, values will be normalised againts the total and |
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| 270 | thus represent frequencies rather than counts. |
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| 271 | """ |
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| 272 | |
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[6158] | 273 | n = num.searchsorted(num.sort(a), bins) |
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| 274 | n = num.concatenate( [n, [len(a)]] ) |
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[5897] | 275 | |
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| 276 | hist = n[1:]-n[:-1] |
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| 277 | |
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| 278 | if relative is True: |
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[6158] | 279 | hist = hist/float(num.sum(hist)) |
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[5897] | 280 | |
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| 281 | return hist |
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| 282 | |
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| 283 | def create_bins(data, number_of_bins = None): |
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| 284 | """Safely create bins for use with histogram |
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| 285 | If data contains only one point or is constant, one bin will be created. |
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| 286 | If number_of_bins in omitted 10 bins will be created |
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| 287 | """ |
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| 288 | |
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| 289 | mx = max(data) |
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| 290 | mn = min(data) |
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| 291 | |
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| 292 | if mx == mn: |
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[6158] | 293 | bins = num.array([mn]) |
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[5897] | 294 | else: |
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| 295 | if number_of_bins is None: |
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| 296 | number_of_bins = 10 |
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| 297 | |
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[6158] | 298 | bins = num.arange(mn, mx, (mx-mn)/number_of_bins) |
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[5897] | 299 | |
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| 300 | return bins |
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| 301 | |
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| 302 | |
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| 303 | |
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| 304 | def get_machine_precision(): |
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| 305 | """Calculate the machine precision for Floats |
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| 306 | |
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| 307 | Depends on static variable machine_precision in this module |
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| 308 | as this would otherwise require too much computation. |
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| 309 | """ |
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| 310 | |
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| 311 | global machine_precision |
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| 312 | |
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| 313 | if machine_precision is None: |
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| 314 | epsilon = 1. |
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| 315 | while epsilon/2 + 1. > 1.: |
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| 316 | epsilon /= 2 |
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| 317 | |
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| 318 | machine_precision = epsilon |
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| 319 | |
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| 320 | return machine_precision |
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| 321 | |
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| 322 | #################################################################### |
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| 323 | #Python versions of function that are also implemented in numerical_tools_ext.c |
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| 324 | # FIXME (Ole): Delete these and update tests |
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| 325 | # |
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| 326 | |
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| 327 | def gradient_python(x0, y0, x1, y1, x2, y2, q0, q1, q2): |
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| 328 | """ |
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| 329 | """ |
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| 330 | |
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| 331 | det = (y2-y0)*(x1-x0) - (y1-y0)*(x2-x0) |
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| 332 | a = (y2-y0)*(q1-q0) - (y1-y0)*(q2-q0) |
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| 333 | a /= det |
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| 334 | |
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| 335 | b = (x1-x0)*(q2-q0) - (x2-x0)*(q1-q0) |
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| 336 | b /= det |
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| 337 | |
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| 338 | return a, b |
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| 339 | |
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| 340 | |
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| 341 | def gradient2_python(x0, y0, x1, y1, q0, q1): |
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| 342 | """Compute radient based on two points and enforce zero gradient |
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| 343 | in the direction orthogonal to (x1-x0), (y1-y0) |
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| 344 | """ |
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| 345 | |
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| 346 | #Old code |
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| 347 | #det = x0*y1 - x1*y0 |
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| 348 | #if det != 0.0: |
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| 349 | # a = (y1*q0 - y0*q1)/det |
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| 350 | # b = (x0*q1 - x1*q0)/det |
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| 351 | |
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| 352 | #Correct code (ON) |
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| 353 | det = (x1-x0)**2 + (y1-y0)**2 |
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| 354 | if det != 0.0: |
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| 355 | a = (x1-x0)*(q1-q0)/det |
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| 356 | b = (y1-y0)*(q1-q0)/det |
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| 357 | |
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| 358 | return a, b |
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| 359 | |
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| 360 | |
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| 361 | #----------------- |
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| 362 | #Initialise module |
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| 363 | |
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| 364 | from anuga.utilities import compile |
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| 365 | if compile.can_use_C_extension('util_ext.c'): |
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| 366 | from util_ext import gradient, gradient2 |
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| 367 | else: |
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| 368 | gradient = gradient_python |
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| 369 | gradient2 = gradient2_python |
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| 370 | |
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| 371 | |
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[6119] | 372 | if __name__ == '__main__': |
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[5897] | 373 | pass |
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| 374 | |
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| 375 | |
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