MATLAB Function Reference

pchip

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)

Syntax

yi = pchip(x,y,xi)
pp = pchip(x,y)

Description

yi = pchip(x,y,xi) returns vector yi containing elements corresponding to the elements of xi and determined by piecewise cubic interpolation within vectors x and y. The vector x specifies the points at which the data y is given. If y is a matrix, then the interpolation is performed for each column of y and yi is length(xi)-by-size(y,2).

pp = pchip(x,y) returns a piecewise polynomial structure for use by ppval. x can be a row or column vector. y is a row or column vector of the same length as x, or a matrix with length(x) columns.

pchip finds values of an underlying interpolating function at intermediate points. satisfies:

• is a different cubic on each subinterval, .
• interpolates the data, i.e. .
• The first derivative, , is continuous.
• The second derivative, , is piecewise linear.
• is probably not continuous; there may be jumps at .
• preserves both the shape of the data and monotonicity.
• On intervals where the data is monotonic, so is .
• At points where the data has a local extremum, so does .

Note    If is a matrix, satisfies the above for each column of .

Comparing pchip with spline:

• spline is smoother, i.e. is continuous.
• spline is more accurate if the data are values of a smooth function.
• pchip has no overshoots and less oscillation if the data are not smooth.
• pchip is less expensive to set up.
• The two are equally expensive to evaluate.

Examples

x = -3:3; 
y = [-1 -1 -1 0 1 1 1]; 
t = -3:.01:3;
p = pchip(x,y,t);
s = spline(x,y,t);
plot(x,y,'o',t,p,'-',t,s,'-.')
legend({'data','pchip','spline'})

See Also

interp1, spline, ppval

References

[1]  Fritsch, F. N. and R. E. Carlson, "Monotone Piecewise Cubic Interpolation," SIAM J. Numerical Analysis, Vol. 17, 1980, pp.238-246.

[2]  Kahaner, David, Cleve Moler, Stephen Nash, Numerical Methods and Software, Prentice Hall, 1988.

pcg

pcode