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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9 | import Numeric as num |
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10 | |
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11 | NAN = (num.array([1])/0.)[0] |
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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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76 | v1 = ensure_numeric(v1, num.Float) |
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77 | v2 = ensure_numeric(v2, num.Float) |
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78 | |
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79 | # Normalise |
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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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82 | |
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83 | # Compute angle |
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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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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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127 | return num.array([-v[1], v[0]], num.Float) |
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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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137 | return(float(num.sum(x))/len(x)) |
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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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158 | p = num.innerproduct(cx,cy)/N |
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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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207 | y = num.ravel(x) |
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208 | p = num.sqrt(num.innerproduct(y,y)) |
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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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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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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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250 | if type(A) == num.ArrayType: |
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251 | return A |
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252 | else: |
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253 | return num.array(A) |
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254 | else: |
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255 | if type(A) == num.ArrayType: |
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256 | if A.typecode == typecode: |
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257 | return A |
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258 | else: |
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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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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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273 | n = num.searchsorted(num.sort(a), bins) |
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274 | n = num.concatenate( [n, [len(a)]] ) |
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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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279 | hist = hist/float(num.sum(hist)) |
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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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293 | bins = num.array([mn]) |
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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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298 | bins = num.arange(mn, mx, (mx-mn)/number_of_bins) |
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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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372 | if __name__ == '__main__': |
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373 | pass |
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374 | |
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375 | |
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