# source:branches/numpy/anuga/utilities/numerical_tools.py@6441

Last change on this file since 6441 was 6441, checked in by rwilson, 14 years ago

After changes to get_absolute, ensure_numeric, etc.

File size: 10.2 KB
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1#!/usr/bin/env python
2"""Auxiliary numerical tools
3
4"""
5
6from math import acos, pi, sqrt
7from warnings import warn
8
9import numpy as num
10
11NAN = (num.array([1])/0.)[0]
12# if we use a package that has NAN, this should be updated to use NAN.
13
14# Static variable used by get_machine_precision
15machine_precision = None
16
17
18def safe_acos(x):
19    """Safely compute acos
20
21    Protect against cases where input argument x is outside the allowed
22    interval [-1.0, 1.0] by no more than machine precision
23    """
24
25    error_msg = 'Input to acos is outside allowed domain [-1.0, 1.0].'+\
26                'I got %.12f' %x
27    warning_msg = 'Changing argument to acos from %.18f to %.1f' %(x, sign(x))
28
29    eps = get_machine_precision() # Machine precision
30    if x < -1.0:
31        if x < -1.0 - eps:
32            raise ValueError, errmsg
33        else:
34            warn(warning_msg)
35            x = -1.0
36
37    if x > 1.0:
38        if x > 1.0 + eps:
39            raise ValueError, errmsg
40        else:
41            print 'NOTE: changing argument to acos from %.18f to 1.0' %x
42            x = 1.0
43
44    return acos(x)
45
46
47def sign(x):
48    if x > 0: return 1
49    if x < 0: return -1
50    if x == 0: return 0
51
52
53def is_scalar(x):
54    """True if x is a scalar (constant numeric value)
55    """
56
57    from types import IntType, FloatType
58    if type(x) in [IntType, FloatType]:
59        return True
60    else:
61        return False
62
63def angle(v1, v2=None):
64    """Compute angle between 2D vectors v1 and v2.
65
66    If v2 is not specified it will default
67    to e1 (the unit vector in the x-direction)
68
69    The angle is measured as a number in [0, 2pi] from v2 to v1.
70    """
71
72    # Prepare two numeric vectors
73    if v2 is None:
74        v2 = [1.0, 0.0] # Unit vector along the x-axis
75
76    v1 = ensure_numeric(v1, num.float)
77    v2 = ensure_numeric(v2, num.float)
78
79    # Normalise
80    v1 = v1/num.sqrt(num.sum(v1**2))
81    v2 = v2/num.sqrt(num.sum(v2**2))
82
83    # Compute angle
84    p = num.inner(v1, v2)
85    c = num.inner(v1, normal_vector(v2)) # Projection onto normal
86                                            # (negative cross product)
87
88    theta = safe_acos(p)
89
90
91    # Correct if v1 is in quadrant 3 or 4 with respect to v2 (as the x-axis)
92    # If v2 was the unit vector [1,0] this would correspond to the test
93    # if v1[1] < 0: theta = 2*pi-theta
94    # In general we use the sign of the projection onto the normal.
95    if c < 0:
97       theta = 2*pi-theta
98
99    return theta
100
101
102def anglediff(v0, v1):
103    """Compute difference between angle of vector v0 (x0, y0) and v1 (x1, y1).
104    This is used for determining the ordering of vertices,
105    e.g. for checking if they are counter clockwise.
106
107    Always return a positive value
108    """
109
110    from math import pi
111
112    a0 = angle(v0)
113    a1 = angle(v1)
114
115    #Ensure that difference will be positive
116    if a0 < a1:
117        a0 += 2*pi
118
119    return a0-a1
120
121def normal_vector(v):
122    """Normal vector to v.
123
124    Returns vector 90 degrees counter clockwise to and of same length as v
125    """
126
127    return num.array([-v[1], v[0]], num.float)
128
129
130#def crossproduct_length(v1, v2):
131#    return v1[0]*v2[1]-v2[0]*v1[1]
132
133
134def mean(x):
135    """Mean value of a vector
136    """
137    return(float(num.sum(x))/len(x))
138
139
140def cov(x, y=None):
141    """Covariance of vectors x and y.
142
143    If y is None: return cov(x, x)
144    """
145
146    if y is None:
147        y = x
148
149    x = ensure_numeric(x)
150    y = ensure_numeric(y)
151    msg = 'Lengths must be equal: len(x) == %d, len(y) == %d' %(len(x), len(y))
152    assert(len(x)==len(y)), msg
153
154    N = len(x)
155    cx = x - mean(x)
156    cy = y - mean(y)
157
158    p = num.inner(cx,cy)/N
159    return(p)
160
161
162def err(x, y=0, n=2, relative=True):
163    """Relative error of ||x-y|| to ||y||
164       n = 2:    Two norm
165       n = None: Max norm
166
167       If denominator evaluates to zero or
168       if y is omitted or
169       if keyword relative is False,
170       absolute error is returned
171
172       If there is x and y, n=2 and relative=False, this will calc;
173       sqrt(sum_over_x&y((xi - yi)^2))
174
175       Given this value (err), to calc the root mean square deviation, do
176       err/sqrt(n)
177       where n is the number of elements,(len(x))
178    """
179
180    x = ensure_numeric(x)
181    if y:
182        y = ensure_numeric(y)
183
184    if n == 2:
185        err = norm(x-y)
186        if relative is True:
187            try:
188                err = err/norm(y)
189            except:
190                pass
191
192    else:
193        err = max(abs(x-y))
194        if relative is True:
195            try:
196                err = err/max(abs(y))
197            except:
198                pass
199
200    return err
201
202
203def norm(x):
204    """2-norm of x
205    """
206
207    y = num.ravel(x)
208    p = num.sqrt(num.inner(y,y))
209    return p
210
211
212def corr(x, y=None):
213    """Correlation of x and y
214    If y is None return autocorrelation of x
215    """
216
217    from math import sqrt
218    if y is None:
219        y = x
220
221    varx = cov(x)
222    vary = cov(y)
223
224    if varx == 0 or vary == 0:
225        C = 0
226    else:
227        C = cov(x,y)/sqrt(varx * vary)
228
229    return(C)
230
231
232
233##
234# @brief Ensure that a sequence is a numeric array of the required type.
235# @param A The sequence object to convert to a numeric array.
236# @param typecode The required numeric type of object A (a numeric dtype).
237# @return A numeric array of the required type.
238def ensure_numeric(A, typecode=None):
239    """Ensure that sequence is a numeric array.
240    Inputs:
241        A: Sequence. If A is already a numeric array it will be returned
242                     unaltered
243                     If not, an attempt is made to convert it to a numeric
244                     array
245        A: Scalar.   Return 0-dimensional array containing that value. Note
246                     that a 0-dim array DOES NOT HAVE A LENGTH UNDER numpy.
247        A: String.   Array of ASCII values (numpy can't handle this)
248
249        typecode:    numeric type. If specified, use this in the conversion.
250                     If not, let numeric package decide.
251                     numpy assumes float64 if no type in A.
252                     typecode will always be one of num.float, num.int, etc.
253
254    Note that num.array(A, dtype) will sometimes copy.  Use 'copy=False' to
255    copy only when required.
256
257    This function is necessary as array(A) can cause memory overflow.
258    """
259
260    if isinstance(A, basestring):
261        msg = 'Sorry, cannot handle string in ensure_numeric()'
262        raise Exception, msg
263
264    if typecode is None:
265        if isinstance(A, num.ndarray):
266            return A
267        else:
268            return num.array(A)
269    else:
270        return num.array(A, dtype=typecode, copy=False)
271
272
273
274
275def histogram(a, bins, relative=False):
276    """Standard histogram straight from the numeric manual
277
278    If relative is True, values will be normalised againts the total and
279    thus represent frequencies rather than counts.
280    """
281
282    n = num.searchsorted(num.sort(a), bins)
283    n = num.concatenate([n, [len(a)]], axis=0)    #??default#
284
285    hist = n[1:]-n[:-1]
286
287    if relative is True:
288        hist = hist/float(num.sum(hist))
289
290    return hist
291
292def create_bins(data, number_of_bins = None):
293    """Safely create bins for use with histogram
294    If data contains only one point or is constant, one bin will be created.
295    If number_of_bins in omitted 10 bins will be created
296    """
297
298    mx = max(data)
299    mn = min(data)
300
301    if mx == mn:
302        bins = num.array([mn])
303    else:
304        if number_of_bins is None:
305            number_of_bins = 10
306
307        bins = num.arange(mn, mx, (mx-mn)/number_of_bins)
308
309    return bins
310
311
312
313def get_machine_precision():
314    """Calculate the machine precision for Floats
315
316    Depends on static variable machine_precision in this module
317    as this would otherwise require too much computation.
318    """
319
320    global machine_precision
321
322    if machine_precision is None:
323        epsilon = 1.
324        while epsilon/2 + 1. > 1.:
325            epsilon /= 2
326
327        machine_precision = epsilon
328
329    return machine_precision
330
331####################################################################
332#Python versions of function that are also implemented in numerical_tools_ext.c
333# FIXME (Ole): Delete these and update tests
334#
335
336def gradient_python(x0, y0, x1, y1, x2, y2, q0, q1, q2):
337    """
338    """
339
340    det = (y2-y0)*(x1-x0) - (y1-y0)*(x2-x0)
341    a = (y2-y0)*(q1-q0) - (y1-y0)*(q2-q0)
342    a /= det
343
344    b = (x1-x0)*(q2-q0) - (x2-x0)*(q1-q0)
345    b /= det
346
347    return a, b
348
349
350def gradient2_python(x0, y0, x1, y1, q0, q1):
352    in the direction orthogonal to (x1-x0), (y1-y0)
353    """
354
355    #Old code
356    #det = x0*y1 - x1*y0
357    #if det != 0.0:
358    #    a = (y1*q0 - y0*q1)/det
359    #    b = (x0*q1 - x1*q0)/det
360
361    #Correct code (ON)
362    det = (x1-x0)**2 + (y1-y0)**2
363    if det != 0.0:
364        a = (x1-x0)*(q1-q0)/det
365        b = (y1-y0)*(q1-q0)/det
366
367    return a, b
368
369################################################################################
370# Decision functions for numeric package objects.
371# It is a little tricky to decide if a numpy thing is of type float.
372# These functions hide numpy-specific details of how we do this.
373################################################################################
374
375##
376# @brief Decide if an object is a numeric package object with datatype of float.
377# @param obj The object to decide on.
378# @return True if 'obj' is a numeric package object, and some sort of float.
379def is_num_float(obj):
380    '''Is an object a numeric package float object?'''
381
382    try:
383        return obj.dtype.char in num.typecodes['Float']
384    except AttributeError:
385        return False
386
387##
388# @brief Decide if an object is a numeric package object with datatype of int.
389# @param obj The object to decide on.
390# @return True if 'obj' is a numeric package object, and some sort of int.
391def is_num_int(obj):
392    '''Is an object a numeric package int object?'''
393
394    try:
395        return obj.dtype.char in num.typecodes['Integer']
396    except AttributeError:
397        return False
398
399
400#-----------------
401#Initialise module
402
403from anuga.utilities import compile
404if compile.can_use_C_extension('util_ext.c'):