diff options
Diffstat (limited to 'numpy/lib/polynomial.py')
| -rw-r--r-- | numpy/lib/polynomial.py | 87 |
1 files changed, 49 insertions, 38 deletions
diff --git a/numpy/lib/polynomial.py b/numpy/lib/polynomial.py index e33bc4e57..ab2febf35 100644 --- a/numpy/lib/polynomial.py +++ b/numpy/lib/polynomial.py @@ -193,7 +193,6 @@ def polyfit(x, y, deg, rcond=None, full=False): Parameters ---------- - x : array_like 1D vector of sample points. y : array_like @@ -213,17 +212,15 @@ def polyfit(x, y, deg, rcond=None, full=False): Returns ------- - coefficients, [residuals, rank, singular_values, rcond] : variable When full=False, only the coefficients are returned, running down the - appropriate colume when y is a 2D array. When full=True, the rank of - the scaled Vandermonde matrix, it's effective rank in light of the rcond + appropriate colume when y is a 2D array. When full=True, the rank of the + scaled Vandermonde matrix, it's effective rank in light of the rcond value, its singular values, and the specified value of rcond are also returned. Warns ----- - RankWarning : if rank is reduced and not full output The warnings can be turned off by: >>> import numpy as np @@ -233,37 +230,35 @@ def polyfit(x, y, deg, rcond=None, full=False): See Also -------- - polyval : computes polynomial values. Notes ----- - If X is a the Vandermonde Matrix computed from x (see http://mathworld.wolfram.com/VandermondeMatrix.html), then the polynomial least squares solution is given by the 'p' in - X*p = y + X*p = y where X.shape is a matrix of dimensions (len(x), deg + 1), p is a vector of dimensions (deg + 1, 1), and y is a vector of dimensions (len(x), 1). This equation can be solved as - p = (XT*X)^-1 * XT * y + p = (XT*X)^-1 * XT * y where XT is the transpose of X and -1 denotes the inverse. However, this method is susceptible to rounding errors and generally the singular value - decomposition of the matrix X is preferred and that is what is done here. The singular - value method takes a paramenter, 'rcond', which sets a limit on the - relative size of the smallest singular value to be used in solving the - equation. This may result in lowering the rank of the Vandermonde matrix, - in which case a RankWarning is issued. If polyfit issues a RankWarning, try - a fit of lower degree or replace x by x - x.mean(), both of which will + decomposition of the matrix X is preferred and that is what is done here. + The singular value method takes a paramenter, 'rcond', which sets a limit on + the relative size of the smallest singular value to be used in solving the + equation. This may result in lowering the rank of the Vandermonde matrix, in + which case a RankWarning is issued. If polyfit issues a RankWarning, try a + fit of lower degree or replace x by x - x.mean(), both of which will generally improve the condition number. The routine already normalizes the vector x by its maximum absolute value to help in this regard. The rcond - parameter can be set to a value smaller than its default, but the - resulting fit may be spurious. The current default value of rcond is len(x)*eps, where + parameter can be set to a value smaller than its default, but the resulting + fit may be spurious. The current default value of rcond is len(x)*eps, where eps is the relative precision of the floating type being used, generally around 1e-7 and 2e-16 for IEEE single and double precision respectively. This value of rcond is fairly conservative but works pretty well when x - @@ -272,18 +267,16 @@ def polyfit(x, y, deg, rcond=None, full=False): DISCLAIMER: Power series fits are full of pitfalls for the unwary once the degree of the fit becomes large or the interval of sample points is badly - centered. The problem is that the powers x**n are generally a poor - basis for the polynomial functions on the sample interval, resulting in a - Vandermonde matrix is ill conditioned and coefficients sensitive to rounding - erros. The computation of the polynomial - values will also sensitive to rounding errors. Consequently, the quality of - the polynomial fit - should be checked against the data whenever the condition number is large. - The quality of polynomial fits *can not* be taken for granted. If all - you want to do is draw a smooth curve through the y values and polyfit is - not doing the job, try centering the sample range or look into - scipy.interpolate, which includes some nice spline fitting functions that - may be of use. + centered. The problem is that the powers x**n are generally a poor basis for + the polynomial functions on the sample interval, resulting in a Vandermonde + matrix is ill conditioned and coefficients sensitive to rounding erros. The + computation of the polynomial values will also sensitive to rounding errors. + Consequently, the quality of the polynomial fit should be checked against + the data whenever the condition number is large. The quality of polynomial + fits *can not* be taken for granted. If all you want to do is draw a smooth + curve through the y values and polyfit is not doing the job, try centering + the sample range or look into scipy.interpolate, which includes some nice + spline fitting functions that may be of use. For more info, see http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html, @@ -344,19 +337,37 @@ def polyfit(x, y, deg, rcond=None, full=False): def polyval(p, x): - """Evaluate the polynomial p at x. If x is a polynomial then composition. + """Evaluate the polynomial p at x. + + If p is of length N, this function returns the value: + + p[0]*(x**N-1) + p[1]*(x**N-2) + ... + p[N-2]*x + p[N-1] + + If x is a sequence then p(x) will be returned for all elements of x. If x is + another polynomial then the composite polynomial p(x) will be returned. - Description: + Parameters + ---------- + p : {array_like, poly1d} + 1D array of polynomial coefficients from highest degree to zero or an + instance of poly1d. + x : {array_like, poly1d} + A number, a 1D array of numbers, or an instance of poly1d. - If p is of length N, this function returns the value: - p[0]*(x**N-1) + p[1]*(x**N-2) + ... + p[N-2]*x + p[N-1] + Returns + ------- + values : {array, poly1d} + If either p or x is an instance of poly1d, then an instance of poly1d is + returned, otherwise a 1D array is returned. In the case where x is a + poly1d, the result is the composition of the two polynomials, i.e., + substitution is used. - x can be a sequence and p(x) will be returned for all elements of x. - or x can be another polynomial and the composite polynomial p(x) will be - returned. + Notes + ----- + Horners method is used to evaluate the polynomial. Even so, for polynomial + if high degree the values may be inaccurate due to rounding errors. Use + carefully. - Notice: This can produce inaccurate results for polynomials with - significant variability. Use carefully. """ p = NX.asarray(p) if isinstance(x, poly1d): |