API Documentation Generated by Endo, 2006-08-14
API Documentation Generated by Endo, 2006-08-14
Copyright 2002 Pearu Peterson all rights reserved, Pearu Peterson <pearu@cens.ioc.ee> Permission to use, modify, and distribute this software is given under the terms of the SciPy (BSD style) license. See LICENSE.txt that came with this distribution for specifics.
NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK.
Pearu Peterson
myasarray = atleast_1d
Evaluate a bivariate B-spline and its derivatives.
Description:
Return a rank-2 array of spline function values (or spline derivative
values) at points given by the cross-product of the rank-1 arrays x and y.
In special cases, return an array or just a float if either x or y or
both are floats.
Inputs:
x, y -- Rank-1 arrays specifying the domain over which to evaluate the
spline or its derivative.
tck -- A sequence of length 5 returned by bisplrep containing the knot
locations, the coefficients, and the degree of the spline:
[tx, ty, c, kx, ky].
dx, dy -- The orders of the partial derivatives in x and y respectively.
Outputs: (vals, )
vals -- The B-pline or its derivative evaluated over the set formed by
the cross-product of x and y.
Find a bivariate B-spline representation of a surface.
Description:
Given a set of data points (x[i], y[i], z[i]) representing a surface
z=f(x,y), compute a B-spline representation of the surface.
Inputs:
x, y, z -- Rank-1 arrays of data points.
w -- Rank-1 array of weights. By default w=ones(len(x)).
xb, xe -- End points of approximation interval in x.
yb, ye -- End points of approximation interval in y.
By default xb, xe, yb, ye = x.min(), x.max(), y.min(), y.max()
kx, ky -- The degrees of the spline (1 <= kx, ky <= 5). Third order
(kx=ky=3) is recommended.
task -- If task=0, find knots in x and y and coefficients for a given
smoothing factor, s.
If task=1, find knots and coefficients for another value of the
smoothing factor, s. bisplrep must have been previously called
with task=0 or task=1.
If task=-1, find coefficients for a given set of knots tx, ty.
s -- A non-negative smoothing factor. If weights correspond
to the inverse of the standard-deviation of the errors in z,
then a good s-value should be found in the range
(m-sqrt(2*m),m+sqrt(2*m)) where m=len(x)
eps -- A threshold for determining the effective rank of an
over-determined linear system of equations (0 < eps < 1)
--- not likely to need changing.
tx, ty -- Rank-1 arrays of the knots of the spline for task=-1
full_output -- Non-zero to return optional outputs.
nxest, nyest -- Over-estimates of the total number of knots.
If None then nxest = max(kx+sqrt(m/2),2*kx+3),
nyest = max(ky+sqrt(m/2),2*ky+3)
quiet -- Non-zero to suppress printing of messages.
Outputs: (tck, {fp, ier, msg})
tck -- A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and
coefficients (c) of the bivariate B-spline representation of the
surface along with the degree of the spline.
fp -- The weighted sum of squared residuals of the spline approximation.
ier -- An integer flag about splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised.
Otherwise an error is raised.
msg -- A message corresponding to the integer flag, ier.
Remarks:
SEE bisplev to evaluate the value of the B-spline given its tck
representation.
Evaluate all derivatives of a B-spline.
Description:
Given the knots and coefficients of a cubic B-spline compute all
derivatives up to order k at a point (or set of points).
Inputs:
tck -- A length 3 sequence describing the given spline (See splev).
x -- A point or a set of points at which to evaluate the derivatives.
Note that t(k) <= x <= t(n-k+1) must hold for each x.
Outputs: (results, )
results -- An array (or a list of arrays) containing all derivatives
up to order k inclusive for each point x.
Evaulate a B-spline and its derivatives.
Description:
Given the knots and coefficients of a B-spline representation, evaluate the value of the smoothing polynomial and it's derivatives. This is a wrapper around the FORTRAN routines splev and splder of FITPACK.
Inputs:
- x (u) -- a 1-D array of points at which to return the value of the
- smoothed spline or its derivatives. If tck was returned from splprep, then the parameter values, u should be given.
- tck -- A sequence of length 3 returned by splrep or splprep containg the
- knots, coefficients, and degree of the spline.
- der -- The order of derivative of the spline to compute (must be less than
- or equal to k).
Outputs: (y, )
- y -- an array of values representing the spline function or curve.
- If tck was returned from splrep, then this is a list of arrays representing the curve in N-dimensional space.
Evaluate the definite integral of a B-spline.
Description:
Given the knots and coefficients of a B-spline, evaluate the definite
integral of the smoothing polynomial between two given points.
Inputs:
a, b -- The end-points of the integration interval.
tck -- A length 3 sequence describing the given spline (See splev).
full_output -- Non-zero to return optional output.
Outputs: (integral, {wrk})
integral -- The resulting integral.
wrk -- An array containing the integrals of the normalized B-splines defined
on the set of knots.
Find the B-spline representation of an N-dimensional curve.
Description:
Given a list of N rank-1 arrays, x, which represent a curve in N-dimensional
space parametrized by u, find a smooth approximating spline curve g(u).
Uses the FORTRAN routine parcur from FITPACK
Inputs:
x -- A list of sample vector arrays representing the curve.
u -- An array of parameter values. If not given, these values are
calculated automatically as (M = len(x[0])):
v[0] = 0
v[i] = v[i-1] + distance(x[i],x[i-1])
u[i] = v[i] / v[M-1]
ub, ue -- The end-points of the parameters interval. Defaults to
u[0] and u[-1].
k -- Degree of the spline. Cubic splines are recommended. Even values of
k should be avoided especially with a small s-value.
1 <= k <= 5.
task -- If task==0 find t and c for a given smoothing factor, s.
If task==1 find t and c for another value of the smoothing factor,
s. There must have been a previous call with task=0 or task=1
for the same set of data.
If task=-1 find the weighted least square spline for a given set of
knots, t.
s -- A smoothing condition. The amount of smoothness is determined by
satisfying the conditions: sum((w * (y - g))**2) <= s where
g(x) is the smoothed interpolation of (x,y). The user can use s to
control the tradeoff between closeness and smoothness of fit. Larger
s means more smoothing while smaller values of s indicate less
smoothing. Recommended values of s depend on the weights, w. If the
weights represent the inverse of the standard-deviation of y, then a
good s value should be found in the range (m-sqrt(2*m),m+sqrt(2*m))
where m is the number of datapoints in x, y, and w.
t -- The knots needed for task=-1.
full_output -- If non-zero, then return optional outputs.
nest -- An over-estimate of the total number of knots of the spline to
help in determining the storage space. By default nest=m/2.
Always large enough is nest=m+k+1.
per -- If non-zero, data points are considered periodic with period
x[m-1] - x[0] and a smooth periodic spline approximation is returned.
Values of y[m-1] and w[m-1] are not used.
quiet -- Non-zero to suppress messages.
Outputs: (tck, u, {fp, ier, msg})
tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
u -- An array of the values of the parameter.
fp -- The weighted sum of squared residuals of the spline approximation.
ier -- An integer flag about splrep success. Success is indicated
if ier<=0. If ier in [1,2,3] an error occurred but was not raised.
Otherwise an error is raised.
msg -- A message corresponding to the integer flag, ier.
Remarks:
SEE splev for evaluation of the spline and its derivatives.
Find the B-spline representation of 1-D curve.
Description:
Given the set of data points (x[i], y[i]) determine a smooth spline
approximation of degree k on the interval xb <= x <= xe. The coefficients,
c, and the knot points, t, are returned. Uses the FORTRAN routine
curfit from FITPACK.
Inputs:
x, y -- The data points defining a curve y = f(x).
w -- Strictly positive rank-1 array of weights the same length as x and y.
The weights are used in computing the weighted least-squares spline
fit. If the errors in the y values have standard-deviation given by the
vector d, then w should be 1/d. Default is ones(len(x)).
xb, xe -- The interval to fit. If None, these default to x[0] and x[-1]
respectively.
k -- The order of the spline fit. It is recommended to use cubic splines.
Even order splines should be avoided especially with small s values.
1 <= k <= 5
task -- If task==0 find t and c for a given smoothing factor, s.
If task==1 find t and c for another value of the
smoothing factor, s. There must have been a previous
call with task=0 or task=1 for the same set of data
(t will be stored an used internally)
If task=-1 find the weighted least square spline for
a given set of knots, t. These should be interior knots
as knots on the ends will be added automatically.
s -- A smoothing condition. The amount of smoothness is determined by
satisfying the conditions: sum((w * (y - g))**2) <= s where
g(x) is the smoothed interpolation of (x,y). The user can use s to
control the tradeoff between closeness and smoothness of fit. Larger
s means more smoothing while smaller values of s indicate less
smoothing. Recommended values of s depend on the weights, w. If the
weights represent the inverse of the standard-deviation of y, then a
good s value should be found in the range (m-sqrt(2*m),m+sqrt(2*m))
where m is the number of datapoints in x, y, and w.
default : s=m-sqrt(2*m)
t -- The knots needed for task=-1. If given then task is automatically
set to -1.
full_output -- If non-zero, then return optional outputs.
per -- If non-zero, data points are considered periodic with period
x[m-1] - x[0] and a smooth periodic spline approximation is returned.
Values of y[m-1] and w[m-1] are not used.
quiet -- Non-zero to suppress messages.
Outputs: (tck, {fp, ier, msg})
tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
fp -- The weighted sum of squared residuals of the spline approximation.
ier -- An integer flag about splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised.
Otherwise an error is raised.
msg -- A message corresponding to the integer flag, ier.
Remarks:
SEE splev for evaluation of the spline and its derivatives.
Find the roots of a cubic B-spline.
Description:
Given the knots (>=8) and coefficients of a cubic B-spline return the
roots of the spline.
Inputs:
tck -- A length 3 sequence describing the given spline (See splev).
The number of knots must be >= 8. The knots must be a montonically
increasing sequence.
mest -- An estimate of the number of zeros (Default is 10).
Outputs: (zeros, )
zeros -- An array giving the roots of the spline.
| Local name | Refers to |
|---|---|
| arange | numpy.arange |
| array | numpy.array |
| atleast_1d | numpy.atleast_1d |
| cos | numpy.cos |
| dfitpack | dfitpack |
| dot | numpy.dot |
| empty | numpy.empty |
| ones | numpy.ones |
| pi | numpy.pi |
| ravel | numpy.ravel |
| sin | numpy.sin |
| sqrt | numpy.sqrt |
| transpose | numpy.transpose |
| zeros | numpy.zeros |
| _fitpack | _fitpack |