Recently, I had a need for a monotonic piece-wise cubic Hermite
interpolator. Matlab provides the function "pchip" (Piecewise Cubic
Hermite Interpolator), but when I Googled I didn't find any Python
equivalent. I tried "interp1d()" from scipy.interpolate but this was
a standard cubic spline using all of the data - not a piece-wise cubic
spline.
I had access to Matlab documentation, so I spent a some time tracing
through the code to figure out how I might write a Python duplicate.
This was an massive exercise in frustration and a potent reminder on
why I love Python and use Matlab only under duress. I find typical
Matlab code is poorly documented (if at all) and that apparently
includes the code included in their official releases. I also find
Matlab syntax "dated" and the code very difficult to "read".
Wikipedia to the rescue.
Not to be deterred, I found a couple of very well written Wikipedia
entries, which explained in simple language how to compute the
interpolant. Hats off to whoever wrote these entries – they are
excellent. The result was a surprising small amount of code
considering the Matlab code was approaching 10 pages of
incomprehensible code. Again - strong evidence that things are just
better in Python...
Offered for those who might have the same need – a Python pchip()
equivalent ==> pypchip(). Since I'm not sure how attachments work (or
if they work at all...), I copied the code I used below, followed by a
PNG showing "success":
#
# pychip.py
# Michalski
# 20090818
#
# Piecewise cubic Hermite interpolation (monotonic...) in Python
#
# References:
#
# Wikipedia: Monotone cubic interpolation
# Cubic Hermite spline
#
# A cubic Hermte spline is a third degree spline with each
polynomial of the spline
# in Hermite form. The Hermite form consists of two control points
and two control
# tangents for each polynomial. Each interpolation is performed on
one sub-interval
# at a time (piece-wise). A monotone cubic interpolation is a
variant of cubic
# interpolation that preserves monotonicity of the data to be
interpolated (in other
# words, it controls overshoot). Monotonicity is preserved by
linear interpolation
# but not by cubic interpolation.
#
# Use:
#
# There are two separate calls, the first call, pchip_init(),
computes the slopes that
# the interpolator needs. If there are a large number of points to
compute,
# it is more efficient to compute the slopes once, rather than for
every point
# being evaluated. The second call, pchip_eval(), takes the slopes
computed by
# pchip_init() along with X, Y, and a vector of desired "xnew"s and
computes a vector
# of "ynew"s. If only a handful of points is needed, pchip() is a
third function
# which combines a call to pchip_init() followed by pchip_eval().
#
import pylab as P
#=========================================================
def pchip(x, y, xnew):
# Compute the slopes used by the piecewise cubic Hermite
interpolator
m = pchip_init(x, y)
# Use these slopes (along with the Hermite basis function) to
interpolate
ynew = pchip_eval(x, y, xnew)
return ynew
#=========================================================
def x_is_okay(x,xvec):
# Make sure "x" and "xvec" satisfy the conditions for
# running the pchip interpolator
n = len(x)
m = len(xvec)
# Make sure "x" is in sorted order (brute force, but works...)
xx = x.copy()
xx.sort()
total_matches = (xx == x).sum()
if total_matches != n:
print "*" * 50
print "x_is_okay()"
print "x values weren't in sorted order --- aborting"
return False
# Make sure 'x' doesn't have any repeated values
delta = x[1:] - x[:-1]
if (delta == 0.0).any():
print "*" * 50
print "x_is_okay()"
print "x values weren't monotonic--- aborting"
return False
# Check for in-range xvec values (beyond upper edge)
check = xvec > x[-1]
if check.any():
print "*" * 50
print "x_is_okay()"
print "Certain 'xvec' values are beyond the upper end of 'x'"
print "x_max = ", x[-1]
indices = P.compress(check, range(m))
print "out-of-range xvec's = ", xvec[indices]
print "out-of-range xvec indices = ", indices
return False
# Second - check for in-range xvec values (beyond lower edge)
check = xvec< x[0]
if check.any():
print "*" * 50
print "x_is_okay()"
print "Certain 'xvec' values are beyond the lower end of 'x'"
print "x_min = ", x[0]
indices = P.compress(check, range(m))
print "out-of-range xvec's = ", xvec[indices]
print "out-of-range xvec indices = ", indices
return False
return True
#=========================================================
def pchip_eval(x, y, m, xvec):
# Evaluate the piecewise cubic Hermite interpolant with
monoticity preserved
#
# x = array containing the x-data
# y = array containing the y-data
# m = slopes at each (x,y) point [computed to preserve
monotonicity]
# xnew = new "x" value where the interpolation is desired
#
# x must be sorted low to high... (no repeats)
# y can have repeated values
#
# This works with either a scalar or vector of "xvec"
n = len(x)
mm = len(xvec)
############################
# Make sure there aren't problems with the input data
############################
if not x_is_okay(x, xvec):
print "pchip_eval2() - ill formed 'x' vector!!!!!!!!!!!!!"
# Cause a hard crash...
STOP_pchip_eval2
# Find the indices "k" such that x[k] < xvec < x[k+1]
# Create "copies" of "x" as rows in a mxn 2-dimensional vector
xx = P.resize(x,(mm,n)).transpose()
xxx = xx > xvec
# Compute column by column differences
z = xxx[:-1,:] - xxx[1:,:]
# Collapse over rows...
k = z.argmax(axis=0)
# Create the Hermite coefficients
h = x[k+1] - x[k]
t = (xvec - x[k]) / h[k]
# Hermite basis functions
h00 = (2 * t**3) - (3 * t**2) + 1
h10 = t**3 - (2 * t**2) + t
h01 = (-2* t**3) + (3 * t**2)
h11 = t**3 - t**2
# Compute the interpolated value of "y"
ynew = h00*y[k] + h10*h*m[k] + h01*y[k+1] + h11*h*m[k+1]
return ynew
#=========================================================
def pchip_init(x,y):
# Evaluate the piecewise cubic Hermite interpolant with
monoticity preserved
#
# x = array containing the x-data
# y = array containing the y-data
#
# x must be sorted low to high... (no repeats)
# y can have repeated values
#
# x input conditioning is assumed but not checked
n = len(x)
# Compute the slopes of the secant lines between successive points
delta = (y[1:] - y[:-1]) / (x[1:] - x[:-1])
# Initialize the tangents at every points as the average of the
secants
m = P.zeros(n, dtype='d')
# At the endpoints - use one-sided differences
m[0] = delta[0]
m[n-1] = delta[-1]
# In the middle - use the average of the secants
m[1:-1] = (delta[:-1] + delta[1:]) / 2.0
# Special case: intervals where y[k] == y[k+1]
# Setting these slopes to zero guarantees the spline connecting
# these points will be flat which preserves monotonicity
indices_to_fix = P.compress((delta == 0.0), range(n))
# print "zero slope indices to fix = ", indices_to_fix
for ii in indices_to_fix:
m[ii] = 0.0
m[ii+1] = 0.0
alpha = m[:-1]/delta
beta = m[1:]/delta
dist = alpha**2 + beta**2
tau = 3.0 / P.sqrt(dist)
# To prevent overshoot or undershoot, restrict the position vector
# (alpha, beta) to a circle of radius 3. If (alpha**2 +
beta**2)>9,
# then set m[k] = tau[k]alpha[k]delta[k] and m[k+1] =
tau[k]beta[b]delta[k]
# where tau = 3/sqrt(alpha**2 + beta**2).
# Find the indices that need adjustment
over = (dist > 9.0)
indices_to_fix = P.compress(over, range(n))
# print "overshoot indices to fix... = ", indices_to_fix
for ii in indices_to_fix:
m[ii] = tau[ii] * alpha[ii] * delta[ii]
m[ii+1] = tau[ii] * beta[ii] * delta[ii]
return m
#=
=======================================================================
def CubicHermiteSpline(x, y, x_new):
# Piecewise Cubic Hermite Interpolation using Catmull-Rom
# method for computing the slopes.
#
# Note - this only works if delta-x is uniform?
# Find the two points which "bracket" "x_new"
found_it = False
for ii in range(len(x)-1):
if (x[ii] <= x_new) and (x[ii+1] > x_new):
found_it = True
break
if not found_it:
print
print "requested x=<%f> outside X range[%f,%f]" % (x_new,
x[0], x[-1])
STOP_CubicHermiteSpline()
# Starting and ending data points
x0 = x[ii]
x1 = x[ii+1]
y0 = y[ii]
y1 = y[ii+1]
# Starting and ending tangents (using Catmull-Rom spline method)
# Handle special cases (hit one of the endpoints...)
if ii == 0:
# Hit lower endpoint
m0 = (y[1] - y[0])
m1 = (y[2] - y[0]) / 2.0
elif ii == (len(x) - 2):
# Hit upper endpoints
m0 = (y[ii+1] - y[ii-1]) / 2.0
m1 = (y[ii+1] - y[ii])
else:
# Inside the field...
m0 = (y[ii+1] - y[ii-1])/ 2.0
m1 = (y[ii+2] - y[ii]) / 2.0
# Normalize to x_new to [0,1] interval
h = (x1 - x0)
t = (x_new - x0) / h
# Compute the four Hermite basis functions
h00 = ( 2.0 * t**3) - (3.0 * t**2) + 1.0
h10 = ( 1.0 * t**3) - (2.0 * t**2) + t
h01 = (-2.0 * t**3) + (3.0 * t**2)
h11 = ( 1.0 * t**3) - (1.0 * t**2)
h = 1
y_new = (h00 * y0) + (h10 * h * m0) + (h01 * y1) + (h11 * h * m1)
return y_new
#==============================================================
def main():
############################################################
# Sine wave test
############################################################
# Create a example vector containing a sine wave.
x = P.arange(30.0)/10.
y = P.sin(x)
# Interpolate the data above to the grid defined by "xvec"
xvec = P.arange(250.)/100.
# Initialize the interpolator slopes
m = pchip_init(x,y)
# Call the monotonic piece-wise Hermite cubic interpolator
yvec2 = pchip_eval(x, y, m, xvec)
P.figure(1)
P.plot(x,y, 'ro')
P.title("pchip() Sin test code")
# Plot the interpolated points
P.plot(xvec, yvec2, 'b')
#########################################################################
# Step function test...
#########################################################################
P.figure(2)
P.title("pchip() step function test")
# Create a step function (will demonstrate monotonicity)
x = P.arange(7.0) - 3.0
y = P.array([-1.0, -1,-1,0,1,1,1])
# Interpolate using monotonic piecewise Hermite cubic spline
xvec = P.arange(599.)/100. - 3.0
# Create the pchip slopes slopes
m = pchip_init(x,y)
# Interpolate...
yvec = pchip_eval(x, y, m, xvec)
# Call the Scipy cubic spline interpolator
from scipy.interpolate import interpolate
function = interpolate.interp1d(x, y, kind='cubic')
yvec2 = function(xvec)
# Non-montonic cubic Hermite spline interpolator using
# Catmul-Rom method for computing slopes...
yvec3 = []
for xx in xvec:
yvec3.append(CubicHermiteSpline(x,y,xx))
yvec3 = P.array(yvec3)
# Plot the results
P.plot(x, y, 'ro')
P.plot(xvec, yvec, 'b')
P.plot(xvec, yvec2, 'k')
P.plot(xvec, yvec3, 'g')
P.xlabel("X")
P.ylabel("Y")
P.title("Comparing pypchip() vs. Scipy interp1d() vs. non-
monotonic CHS")
legends = ["Data", "pypchip()", "interp1d","CHS"]
P.legend(legends, loc="upper left")
P.show()
###################################################################
main()
Chris Michalski wrote: > > Offered for those who might have the same need – a Python pchip() > equivalent ==> pypchip(). Since I'm not sure how attachments work (or > if they work at all...), I copied the code I used below, followed by a > PNG showing "success": Chris, This looks interesting. I successfully ran your program by using copy and paste to get it into a file, but for the future I certainly recommend that you attach such a file directly--file attachments generally work very well these days, but bad things can happen to code included as inline text. It would also be helpful if you include the actual URLs of the Wikipedia articles that you used. (Or did I miss them?) Thanks for providing this code, example, and references. Eric
On Sat, Aug 29, 2009 at 1:12 PM, Eric Firing<ef...@ha...> wrote: > This looks interesting. I successfully ran your program by using copy > and paste to get it into a file, but for the future I certainly > recommend that you attach such a file directly--file attachments > generally work very well these days, but bad things can happen to code > included as inline text. The other thing I recommend is do not use the pylab namespace for any of the numerics. pylab is getting all the numerical functions from numpy, so if you import numpy as np and then refer to any numerical functions you need as np.somefunc. Finally, for the functions to be suitable for inclusion in a production package like numpy or matplotlib.mlab, you should not use any print statements in the function, but rather a combination of warnings.warn or exceptions or if it for matplotlib, use the verbose.report infrastructure. That way users can configure how much verbosity they want, where the output should be directed, etc. After a cleanup, you may want to check with numpy or scipy to see of it could find a home there. There was a discussion at scipy on the need to improve scipy.interpolate and this seems to go part of the way toward that objective. So I would start there. JDH
Thanks for the inputs... perhaps it will provide the impetus for future postings as well... chris On Aug 29, 2009, at 11:49 AM, John Hunter wrote: > On Sat, Aug 29, 2009 at 1:12 PM, Eric Firing<ef...@ha...> > wrote: > >> This looks interesting. I successfully ran your program by using >> copy >> and paste to get it into a file, but for the future I certainly >> recommend that you attach such a file directly--file attachments >> generally work very well these days, but bad things can happen to >> code >> included as inline text. I haven't contributed to matplotlib or numpy even though I've used them for some years now, so I wasn't sure about the "etiquette" of file attachments. The other thing I recommend is do not use the pylab namespace for any > > of the numerics. pylab is getting all the numerical functions from > numpy, so if you > > import numpy as np > > and then refer to any numerical functions you need as np.somefunc. Point well taken. Since pylab exposes most of the numpy calls I use, I typically include pylab instead for nump. > > Finally, for the functions to be suitable for inclusion in a > production package like numpy or matplotlib.mlab, you should not use > any print statements in the function, but rather a combination of > warnings.warn or exceptions or if it for matplotlib, use the > verbose.report infrastructure. That way users can configure how much > verbosity they want, where the output should be directed, etc. Point also well taken. I figured out when there were problems, but even after 7 years of writing large Python package, I haven't found the best way to handle exceptions. Usually I purposely cause a "crash" so I don't miss the fact that the code had ill formed data. > > After a cleanup, you may want to check with numpy or scipy to see of > it could find a home there. There was a discussion at scipy on the > need to improve scipy.interpolate and this seems to go part of the way > toward that objective. So I would start there. I'll send it along to the scipy people. I figured since I figured out a relatively simple solution to a problem that is often encountered, it might find use even in its primitive form. I'll add the URLs to the WIkipedia references as well. > > JDH
Chris Michalski wrote: > Thanks for the inputs... perhaps it will provide the impetus for > future postings as well... I think this would be a great addition to scipy.interpolate. I encourage you to massage your code to fit the API and scipy standards and contribute it. -Chris -- Christopher Barker, Ph.D. Oceanographer Emergency Response Division NOAA/NOS/OR&R (206) 526-6959 voice 7600 Sand Point Way NE (206) 526-6329 fax Seattle, WA 98115 (206) 526-6317 main reception Chr...@no...
Chris,
regarding the print statements in the function you may want to use something
like:
total_matches = (xx == x).sum()
if total_matches != n:
raise ValueError("x values weren't in sorted order")
I also have one question:
in function "pchip_eval" instead of " t = (xvec - x[k]) / h[k]" should not
be "t = (xvec - x[k]) / h" ?
Best regards,
Ricardo Bessa
Chris Michalski-3 wrote:
>
> Recently, I had a need for a monotonic piece-wise cubic Hermite
> interpolator. Matlab provides the function "pchip" (Piecewise Cubic
> Hermite Interpolator), but when I Googled I didn't find any Python
> equivalent. I tried "interp1d()" from scipy.interpolate but this was
> a standard cubic spline using all of the data - not a piece-wise cubic
> spline.
>
> I had access to Matlab documentation, so I spent a some time tracing
> through the code to figure out how I might write a Python duplicate.
> This was an massive exercise in frustration and a potent reminder on
> why I love Python and use Matlab only under duress. I find typical
> Matlab code is poorly documented (if at all) and that apparently
> includes the code included in their official releases. I also find
> Matlab syntax "dated" and the code very difficult to "read".
>
> Wikipedia to the rescue.
>
> Not to be deterred, I found a couple of very well written Wikipedia
> entries, which explained in simple language how to compute the
> interpolant. Hats off to whoever wrote these entries – they are
> excellent. The result was a surprising small amount of code
> considering the Matlab code was approaching 10 pages of
> incomprehensible code. Again - strong evidence that things are just
> better in Python...
>
> Offered for those who might have the same need – a Python pchip()
> equivalent ==> pypchip(). Since I'm not sure how attachments work (or
> if they work at all...), I copied the code I used below, followed by a
> PNG showing "success":
>
> #
> # pychip.py
> # Michalski
> # 20090818
> #
> # Piecewise cubic Hermite interpolation (monotonic...) in Python
> #
> # References:
> #
> # Wikipedia: Monotone cubic interpolation
> # Cubic Hermite spline
> #
> # A cubic Hermte spline is a third degree spline with each
> polynomial of the spline
> # in Hermite form. The Hermite form consists of two control points
> and two control
> # tangents for each polynomial. Each interpolation is performed on
> one sub-interval
> # at a time (piece-wise). A monotone cubic interpolation is a
> variant of cubic
> # interpolation that preserves monotonicity of the data to be
> interpolated (in other
> # words, it controls overshoot). Monotonicity is preserved by
> linear interpolation
> # but not by cubic interpolation.
> #
> # Use:
> #
> # There are two separate calls, the first call, pchip_init(),
> computes the slopes that
> # the interpolator needs. If there are a large number of points to
> compute,
> # it is more efficient to compute the slopes once, rather than for
> every point
> # being evaluated. The second call, pchip_eval(), takes the slopes
> computed by
> # pchip_init() along with X, Y, and a vector of desired "xnew"s and
> computes a vector
> # of "ynew"s. If only a handful of points is needed, pchip() is a
> third function
> # which combines a call to pchip_init() followed by pchip_eval().
> #
>
> import pylab as P
>
> #=========================================================
> def pchip(x, y, xnew):
>
> # Compute the slopes used by the piecewise cubic Hermite
> interpolator
> m = pchip_init(x, y)
>
> # Use these slopes (along with the Hermite basis function) to
> interpolate
> ynew = pchip_eval(x, y, xnew)
>
> return ynew
>
> #=========================================================
> def x_is_okay(x,xvec):
>
> # Make sure "x" and "xvec" satisfy the conditions for
> # running the pchip interpolator
>
> n = len(x)
> m = len(xvec)
>
> # Make sure "x" is in sorted order (brute force, but works...)
> xx = x.copy()
> xx.sort()
> total_matches = (xx == x).sum()
> if total_matches != n:
> print "*" * 50
> print "x_is_okay()"
> print "x values weren't in sorted order --- aborting"
> return False
>
> # Make sure 'x' doesn't have any repeated values
> delta = x[1:] - x[:-1]
> if (delta == 0.0).any():
> print "*" * 50
> print "x_is_okay()"
> print "x values weren't monotonic--- aborting"
> return False
>
> # Check for in-range xvec values (beyond upper edge)
> check = xvec > x[-1]
> if check.any():
> print "*" * 50
> print "x_is_okay()"
> print "Certain 'xvec' values are beyond the upper end of 'x'"
> print "x_max = ", x[-1]
> indices = P.compress(check, range(m))
> print "out-of-range xvec's = ", xvec[indices]
> print "out-of-range xvec indices = ", indices
> return False
>
> # Second - check for in-range xvec values (beyond lower edge)
> check = xvec< x[0]
> if check.any():
> print "*" * 50
> print "x_is_okay()"
> print "Certain 'xvec' values are beyond the lower end of 'x'"
> print "x_min = ", x[0]
> indices = P.compress(check, range(m))
> print "out-of-range xvec's = ", xvec[indices]
> print "out-of-range xvec indices = ", indices
> return False
>
> return True
>
> #=========================================================
> def pchip_eval(x, y, m, xvec):
>
> # Evaluate the piecewise cubic Hermite interpolant with
> monoticity preserved
> #
> # x = array containing the x-data
> # y = array containing the y-data
> # m = slopes at each (x,y) point [computed to preserve
> monotonicity]
> # xnew = new "x" value where the interpolation is desired
> #
> # x must be sorted low to high... (no repeats)
> # y can have repeated values
> #
> # This works with either a scalar or vector of "xvec"
>
> n = len(x)
> mm = len(xvec)
>
> ############################
> # Make sure there aren't problems with the input data
> ############################
> if not x_is_okay(x, xvec):
> print "pchip_eval2() - ill formed 'x' vector!!!!!!!!!!!!!"
>
> # Cause a hard crash...
> STOP_pchip_eval2
>
> # Find the indices "k" such that x[k] < xvec < x[k+1]
>
> # Create "copies" of "x" as rows in a mxn 2-dimensional vector
> xx = P.resize(x,(mm,n)).transpose()
> xxx = xx > xvec
>
> # Compute column by column differences
> z = xxx[:-1,:] - xxx[1:,:]
>
> # Collapse over rows...
> k = z.argmax(axis=0)
>
> # Create the Hermite coefficients
> h = x[k+1] - x[k]
> t = (xvec - x[k]) / h[k]
>
> # Hermite basis functions
> h00 = (2 * t**3) - (3 * t**2) + 1
> h10 = t**3 - (2 * t**2) + t
> h01 = (-2* t**3) + (3 * t**2)
> h11 = t**3 - t**2
>
> # Compute the interpolated value of "y"
> ynew = h00*y[k] + h10*h*m[k] + h01*y[k+1] + h11*h*m[k+1]
>
> return ynew
>
> #=========================================================
> def pchip_init(x,y):
>
> # Evaluate the piecewise cubic Hermite interpolant with
> monoticity preserved
> #
> # x = array containing the x-data
> # y = array containing the y-data
> #
> # x must be sorted low to high... (no repeats)
> # y can have repeated values
> #
> # x input conditioning is assumed but not checked
>
> n = len(x)
>
> # Compute the slopes of the secant lines between successive points
> delta = (y[1:] - y[:-1]) / (x[1:] - x[:-1])
>
> # Initialize the tangents at every points as the average of the
> secants
> m = P.zeros(n, dtype='d')
>
> # At the endpoints - use one-sided differences
> m[0] = delta[0]
> m[n-1] = delta[-1]
>
> # In the middle - use the average of the secants
> m[1:-1] = (delta[:-1] + delta[1:]) / 2.0
>
> # Special case: intervals where y[k] == y[k+1]
>
> # Setting these slopes to zero guarantees the spline connecting
> # these points will be flat which preserves monotonicity
> indices_to_fix = P.compress((delta == 0.0), range(n))
>
> # print "zero slope indices to fix = ", indices_to_fix
>
> for ii in indices_to_fix:
> m[ii] = 0.0
> m[ii+1] = 0.0
>
> alpha = m[:-1]/delta
> beta = m[1:]/delta
> dist = alpha**2 + beta**2
> tau = 3.0 / P.sqrt(dist)
>
> # To prevent overshoot or undershoot, restrict the position vector
> # (alpha, beta) to a circle of radius 3. If (alpha**2 +
> beta**2)>9,
> # then set m[k] = tau[k]alpha[k]delta[k] and m[k+1] =
> tau[k]beta[b]delta[k]
> # where tau = 3/sqrt(alpha**2 + beta**2).
>
> # Find the indices that need adjustment
> over = (dist > 9.0)
> indices_to_fix = P.compress(over, range(n))
>
> # print "overshoot indices to fix... = ", indices_to_fix
>
> for ii in indices_to_fix:
> m[ii] = tau[ii] * alpha[ii] * delta[ii]
> m[ii+1] = tau[ii] * beta[ii] * delta[ii]
>
> return m
>
> #=
> =======================================================================
> def CubicHermiteSpline(x, y, x_new):
>
> # Piecewise Cubic Hermite Interpolation using Catmull-Rom
> # method for computing the slopes.
> #
> # Note - this only works if delta-x is uniform?
>
> # Find the two points which "bracket" "x_new"
> found_it = False
>
> for ii in range(len(x)-1):
> if (x[ii] <= x_new) and (x[ii+1] > x_new):
> found_it = True
> break
>
> if not found_it:
> print
> print "requested x=<%f> outside X range[%f,%f]" % (x_new,
> x[0], x[-1])
> STOP_CubicHermiteSpline()
>
> # Starting and ending data points
> x0 = x[ii]
> x1 = x[ii+1]
>
> y0 = y[ii]
> y1 = y[ii+1]
>
> # Starting and ending tangents (using Catmull-Rom spline method)
>
> # Handle special cases (hit one of the endpoints...)
> if ii == 0:
>
> # Hit lower endpoint
> m0 = (y[1] - y[0])
> m1 = (y[2] - y[0]) / 2.0
>
> elif ii == (len(x) - 2):
>
> # Hit upper endpoints
> m0 = (y[ii+1] - y[ii-1]) / 2.0
> m1 = (y[ii+1] - y[ii])
>
> else:
>
> # Inside the field...
> m0 = (y[ii+1] - y[ii-1])/ 2.0
> m1 = (y[ii+2] - y[ii]) / 2.0
>
> # Normalize to x_new to [0,1] interval
> h = (x1 - x0)
> t = (x_new - x0) / h
>
> # Compute the four Hermite basis functions
> h00 = ( 2.0 * t**3) - (3.0 * t**2) + 1.0
> h10 = ( 1.0 * t**3) - (2.0 * t**2) + t
> h01 = (-2.0 * t**3) + (3.0 * t**2)
> h11 = ( 1.0 * t**3) - (1.0 * t**2)
>
> h = 1
>
> y_new = (h00 * y0) + (h10 * h * m0) + (h01 * y1) + (h11 * h * m1)
>
> return y_new
>
> #==============================================================
> def main():
>
> ############################################################
> # Sine wave test
> ############################################################
>
> # Create a example vector containing a sine wave.
> x = P.arange(30.0)/10.
> y = P.sin(x)
>
> # Interpolate the data above to the grid defined by "xvec"
> xvec = P.arange(250.)/100.
>
> # Initialize the interpolator slopes
> m = pchip_init(x,y)
>
> # Call the monotonic piece-wise Hermite cubic interpolator
> yvec2 = pchip_eval(x, y, m, xvec)
>
> P.figure(1)
> P.plot(x,y, 'ro')
> P.title("pchip() Sin test code")
>
> # Plot the interpolated points
> P.plot(xvec, yvec2, 'b')
>
>
> #########################################################################
> # Step function test...
>
> #########################################################################
>
> P.figure(2)
> P.title("pchip() step function test")
>
> # Create a step function (will demonstrate monotonicity)
> x = P.arange(7.0) - 3.0
> y = P.array([-1.0, -1,-1,0,1,1,1])
>
> # Interpolate using monotonic piecewise Hermite cubic spline
> xvec = P.arange(599.)/100. - 3.0
>
> # Create the pchip slopes slopes
> m = pchip_init(x,y)
>
> # Interpolate...
> yvec = pchip_eval(x, y, m, xvec)
>
> # Call the Scipy cubic spline interpolator
> from scipy.interpolate import interpolate
> function = interpolate.interp1d(x, y, kind='cubic')
> yvec2 = function(xvec)
>
> # Non-montonic cubic Hermite spline interpolator using
> # Catmul-Rom method for computing slopes...
> yvec3 = []
> for xx in xvec:
> yvec3.append(CubicHermiteSpline(x,y,xx))
> yvec3 = P.array(yvec3)
>
> # Plot the results
> P.plot(x, y, 'ro')
> P.plot(xvec, yvec, 'b')
> P.plot(xvec, yvec2, 'k')
> P.plot(xvec, yvec3, 'g')
> P.xlabel("X")
> P.ylabel("Y")
> P.title("Comparing pypchip() vs. Scipy interp1d() vs. non-
> monotonic CHS")
> legends = ["Data", "pypchip()", "interp1d","CHS"]
> P.legend(legends, loc="upper left")
>
> P.show()
>
> ###################################################################
> main()
>
>
>
>
>
>
>
>
>
>
> ------------------------------------------------------------------------------
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