[2526] | 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 | |
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[2531] | 7 | #Establish which Numeric package to use |
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| 8 | #(this should move to somewhere central) |
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| 9 | try: |
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[2704] | 10 | from scipy import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float |
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[2531] | 11 | except: |
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[2633] | 12 | #print 'Could not find scipy - using Numeric' |
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[2704] | 13 | from Numeric import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float |
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[2526] | 14 | |
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[2704] | 15 | # Getting an infinite number to use when using Numeric |
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[2573] | 16 | INF = (array([1])/0.)[0] |
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| 17 | |
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[2526] | 18 | |
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| 19 | |
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[2704] | 20 | def angle(v1, v2=None): |
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| 21 | """Compute angle between 2D vectors v1 and v2. |
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| 22 | |
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| 23 | If v2 is not specified it will default |
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| 24 | to e1 (the unit vector in the x-direction) |
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[2526] | 25 | |
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[2704] | 26 | The angle is measured as a number in [0, 2pi] from v2 to v1. |
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| 27 | """ |
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| 28 | from math import acos, pi, sqrt |
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| 29 | |
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| 30 | # Prepare two Numeric vectors |
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| 31 | if v2 is None: |
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| 32 | v2 = [1.0, 0.0] # Unit vector along the x-axis |
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| 33 | |
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| 34 | v1 = ensure_numeric(v1, Float) |
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| 35 | v2 = ensure_numeric(v2, Float) |
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[2526] | 36 | |
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[2704] | 37 | # Normalise |
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| 38 | v1 = v1/sqrt(sum(v1**2)) |
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| 39 | v2 = v2/sqrt(sum(v2**2)) |
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| 40 | |
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| 41 | # Compute angle |
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| 42 | p = innerproduct(v1, v2) |
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| 43 | c = crossproduct_length(v1, v2) |
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| 44 | |
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| 45 | theta = acos(p) |
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| 46 | |
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| 47 | # Correct if v1 is in quadrant 3 or 4 with respect to v2 (as the x-axis) |
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| 48 | # If v2 was the unit vector [1,0] this would correspond to the test |
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| 49 | # if v1[1] < 0: theta = 2*pi-theta |
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| 50 | # In general we use the sign of the cross product length |
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| 51 | if c > 0: |
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| 52 | #Quadrant 3 or 4 |
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| 53 | theta = 2*pi-theta |
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| 54 | |
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[2526] | 55 | return theta |
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| 56 | |
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[2704] | 57 | |
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[2526] | 58 | def anglediff(v0, v1): |
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[2709] | 59 | """Compute difference between angle of vector v0 (x0, y0) and v1 (x1, y1). |
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[2526] | 60 | This is used for determining the ordering of vertices, |
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| 61 | e.g. for checking if they are counter clockwise. |
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| 62 | |
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| 63 | Always return a positive value |
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| 64 | """ |
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| 65 | |
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| 66 | from math import pi |
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| 67 | |
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| 68 | a0 = angle(v0) |
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| 69 | a1 = angle(v1) |
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| 70 | |
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| 71 | #Ensure that difference will be positive |
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| 72 | if a0 < a1: |
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| 73 | a0 += 2*pi |
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| 74 | |
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| 75 | return a0-a1 |
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| 76 | |
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[2704] | 77 | def normal_vector(v): |
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| 78 | """Normal vector to v |
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| 79 | """ |
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| 80 | |
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| 81 | return array([-v[1], v[0]], Float) |
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[2526] | 82 | |
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[2704] | 83 | |
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| 84 | def crossproduct_length(v1, v2): |
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| 85 | return v1[0]*v2[1]-v2[0]*v1[1] |
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| 86 | |
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| 87 | |
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[2526] | 88 | def mean(x): |
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| 89 | """Mean value of a vector |
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| 90 | """ |
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[2531] | 91 | return(float(sum(x))/len(x)) |
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[2526] | 92 | |
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| 93 | |
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| 94 | def cov(x, y=None): |
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| 95 | """Covariance of vectors x and y. |
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| 96 | |
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| 97 | If y is None: return cov(x, x) |
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| 98 | """ |
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| 99 | |
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| 100 | if y is None: |
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| 101 | y = x |
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| 102 | |
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| 103 | assert(len(x)==len(y)) |
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| 104 | N = len(x) |
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| 105 | |
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| 106 | cx = x - mean(x) |
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| 107 | cy = y - mean(y) |
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| 108 | |
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[2531] | 109 | p = innerproduct(cx,cy)/N |
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[2526] | 110 | return(p) |
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| 111 | |
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| 112 | |
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| 113 | def err(x, y=0, n=2, relative=True): |
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| 114 | """Relative error of ||x-y|| to ||y|| |
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| 115 | n = 2: Two norm |
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| 116 | n = None: Max norm |
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| 117 | |
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| 118 | If denominator evaluates to zero or if y is omitted, |
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| 119 | absolute error is returned |
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| 120 | """ |
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| 121 | |
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| 122 | x = ensure_numeric(x) |
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| 123 | if y: |
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| 124 | y = ensure_numeric(y) |
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| 125 | |
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| 126 | if n == 2: |
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| 127 | err = norm(x-y) |
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| 128 | if relative is True: |
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| 129 | try: |
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| 130 | err = err/norm(y) |
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| 131 | except: |
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| 132 | pass |
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| 133 | |
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| 134 | else: |
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| 135 | err = max(abs(x-y)) |
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| 136 | if relative is True: |
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| 137 | try: |
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| 138 | err = err/max(abs(y)) |
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| 139 | except: |
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| 140 | pass |
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| 141 | |
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| 142 | return err |
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| 143 | |
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| 144 | |
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| 145 | def norm(x): |
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| 146 | """2-norm of x |
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| 147 | """ |
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| 148 | |
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[2531] | 149 | y = ravel(x) |
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| 150 | p = sqrt(innerproduct(y,y)) |
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[2526] | 151 | return p |
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| 152 | |
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| 153 | |
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| 154 | def corr(x, y=None): |
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| 155 | """Correlation of x and y |
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| 156 | If y is None return autocorrelation of x |
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| 157 | """ |
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| 158 | |
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| 159 | from math import sqrt |
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| 160 | if y is None: |
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| 161 | y = x |
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| 162 | |
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| 163 | varx = cov(x) |
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| 164 | vary = cov(y) |
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| 165 | |
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| 166 | if varx == 0 or vary == 0: |
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| 167 | C = 0 |
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| 168 | else: |
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| 169 | C = cov(x,y)/sqrt(varx * vary) |
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| 170 | |
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| 171 | return(C) |
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| 172 | |
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| 173 | |
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| 174 | |
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| 175 | def ensure_numeric(A, typecode = None): |
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| 176 | """Ensure that sequence is a Numeric array. |
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| 177 | Inputs: |
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| 178 | A: Sequence. If A is already a Numeric array it will be returned |
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| 179 | unaltered |
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| 180 | If not, an attempt is made to convert it to a Numeric |
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| 181 | array |
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| 182 | typecode: Numeric type. If specified, use this in the conversion. |
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| 183 | If not, let Numeric decide |
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| 184 | |
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| 185 | This function is necessary as array(A) can cause memory overflow. |
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| 186 | """ |
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| 187 | |
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| 188 | if typecode is None: |
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| 189 | if type(A) == ArrayType: |
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| 190 | return A |
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| 191 | else: |
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| 192 | return array(A) |
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| 193 | else: |
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| 194 | if type(A) == ArrayType: |
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| 195 | if A.typecode == typecode: |
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| 196 | return array(A) #FIXME: Shouldn't this just return A? |
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| 197 | else: |
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| 198 | return A.astype(typecode) |
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| 199 | else: |
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| 200 | return array(A).astype(typecode) |
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| 201 | |
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| 202 | |
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| 203 | |
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[2533] | 204 | |
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| 205 | def histogram(a, bins): |
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| 206 | """Standard histogram straight from the Numeric manual |
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| 207 | """ |
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| 208 | |
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| 209 | n = searchsorted(sort(a), bins) |
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| 210 | n = concatenate( [n, [len(a)]] ) |
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| 211 | return n[1:]-n[:-1] |
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| 212 | |
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| 213 | |
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| 214 | |
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[2526] | 215 | #################################################################### |
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| 216 | #Python versions of function that are also implemented in numerical_tools_ext.c |
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| 217 | # |
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| 218 | |
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| 219 | def gradient_python(x0, y0, x1, y1, x2, y2, q0, q1, q2): |
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| 220 | """ |
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| 221 | """ |
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| 222 | |
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| 223 | det = (y2-y0)*(x1-x0) - (y1-y0)*(x2-x0) |
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| 224 | a = (y2-y0)*(q1-q0) - (y1-y0)*(q2-q0) |
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| 225 | a /= det |
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| 226 | |
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| 227 | b = (x1-x0)*(q2-q0) - (x2-x0)*(q1-q0) |
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| 228 | b /= det |
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| 229 | |
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| 230 | return a, b |
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| 231 | |
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| 232 | |
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| 233 | def gradient2_python(x0, y0, x1, y1, q0, q1): |
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| 234 | """Compute radient based on two points and enforce zero gradient |
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| 235 | in the direction orthogonal to (x1-x0), (y1-y0) |
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| 236 | """ |
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| 237 | |
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| 238 | #Old code |
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| 239 | #det = x0*y1 - x1*y0 |
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| 240 | #if det != 0.0: |
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| 241 | # a = (y1*q0 - y0*q1)/det |
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| 242 | # b = (x0*q1 - x1*q0)/det |
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| 243 | |
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| 244 | #Correct code (ON) |
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| 245 | det = (x1-x0)**2 + (y1-y0)**2 |
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| 246 | if det != 0.0: |
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| 247 | a = (x1-x0)*(q1-q0)/det |
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| 248 | b = (y1-y0)*(q1-q0)/det |
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| 249 | |
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| 250 | return a, b |
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| 251 | |
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| 252 | |
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| 253 | ############################################## |
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| 254 | #Initialise module |
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| 255 | |
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| 256 | from utilities import compile |
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| 257 | if compile.can_use_C_extension('util_ext.c'): |
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| 258 | from util_ext import gradient, gradient2 |
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| 259 | else: |
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| 260 | gradient = gradient_python |
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| 261 | gradient2 = gradient2_python |
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| 262 | |
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| 263 | |
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| 264 | if __name__ == "__main__": |
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| 265 | pass |
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| 266 | |
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[2704] | 267 | |
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| 268 | |
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| 269 | def angle_obsolete(v): |
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| 270 | """Compute angle between e1 (the unit vector in the x-direction) |
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| 271 | and the specified vector v. |
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| 272 | |
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| 273 | Return a number in [0, 2pi] |
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| 274 | """ |
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| 275 | from math import acos, pi, sqrt |
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| 276 | |
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| 277 | # Normalise v |
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| 278 | v = ensure_numeric(v, Float) |
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| 279 | v = v/sqrt(sum(v**2)) |
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| 280 | |
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| 281 | # Compute angle |
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| 282 | theta = acos(v[0]) |
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| 283 | |
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| 284 | if v[1] < 0: |
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| 285 | #Quadrant 3 or 4 |
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| 286 | theta = 2*pi-theta |
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| 287 | |
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| 288 | return theta |
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| 289 | |
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