Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (2024)

Table of Contents
Syntax Description Examples Data Interpolation with spline, pchip, and makima Interpolation with Piecewise Polynomial Structure Input Arguments x — Sample points vector y — Function values at sample points vector | matrix | array xq — Query points scalar | vector | matrix | array Output Arguments p — Interpolated values at query points scalar | vector | matrix | array pp — Piecewise polynomial structure More About Shape-Preserving Piecewise Cubic Interpolation Tips References Extended Capabilities C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™. Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool. GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. Distributed ArraysPartition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™. Version History See Also MATLAB Command Americas Europe Asia Pacific

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)

collapse all in page

Syntax

p = pchip(x,y,xq)

pp = pchip(x,y)

Description

example

p = pchip(x,y,xq) returnsa vector of interpolated values p correspondingto the query points in xq. The values of p aredetermined by shape-preserving piecewise cubicinterpolation of x and y.

example

pp = pchip(x,y) returnsa piecewise polynomial structure for use with ppval andthe spline utility unmkpp.

Examples

collapse all

Data Interpolation with spline, pchip, and makima

Open Live Script

Compare the interpolation results produced by spline, pchip, and makima for two different data sets. These functions all perform different forms of piecewise cubic Hermite interpolation. Each function differs in how it computes the slopes of the interpolant, leading to different behaviors when the underlying data has flat areas or undulations.

Compare the interpolation results on sample data that connects flat regions. Create vectors of x values, function values at those points y, and query points xq. Compute interpolations at the query points using spline, pchip, and makima. Plot the interpolated function values at the query points for comparison.

x = -3:3; y = [-1 -1 -1 0 1 1 1]; xq1 = -3:.01:3;p = pchip(x,y,xq1);s = spline(x,y,xq1);m = makima(x,y,xq1);plot(x,y,'o',xq1,p,'-',xq1,s,'-.',xq1,m,'--')legend('Sample Points','pchip','spline','makima','Location','SouthEast')

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (1)

In this case, pchip and makima have similar behavior in that they avoid overshoots and can accurately connect the flat regions.

Perform a second comparison using an oscillatory sample function.

x = 0:15;y = besselj(1,x);xq2 = 0:0.01:15;p = pchip(x,y,xq2);s = spline(x,y,xq2);m = makima(x,y,xq2);plot(x,y,'o',xq2,p,'-',xq2,s,'-.',xq2,m,'--')legend('Sample Points','pchip','spline','makima')

When the underlying function is oscillatory, spline and makima capture the movement between points better than pchip, which is aggressively flattened near local extrema.

Interpolation with Piecewise Polynomial Structure

Open Live Script

Create vectors for the x values and function values y, and then use pchip to construct a piecewise polynomial structure.

x = -5:5;y = [1 1 1 1 0 0 1 2 2 2 2];p = pchip(x,y);

Use the structure with ppval to evaluate the interpolation at several query points. Plot the results.

xq = -5:0.2:5;pp = ppval(p,xq);plot(x,y,'o',xq,pp,'-.')ylim([-0.2 2.2])

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (3)

Input Arguments

collapse all

x — Sample points
vector

Sample points, specified as a vector. The vector x specifiesthe points at which the data y is given. The elementsof x must be unique.

Data Types: single | double

y — Function values at sample points
vector | matrix | array

Function values at sample points, specified as a numeric vector,matrix, or array. x and y musthave the same length.

If y is a matrix or array, then the valuesin the last dimension, y(:,...,:,j), are takenas the values to match with x. In that case, thelast dimension of y must be the same length as x.

Data Types: single | double

xq — Query points
scalar | vector | matrix | array

Query points, specified as a scalar, vector, matrix, or array. The points specified in xq are the x-coordinates for the interpolated function values yq computed by pchip.

Data Types: single | double

See Also
Conditionally defined expression or function

Output Arguments

collapse all

p — Interpolated values at query points
scalar | vector | matrix | array

Interpolated values at query points, returned as a scalar, vector, matrix, or array. The size of p is related to the sizes of y and xq:

  • If y is a vector, then p has the same size as xq.

  • If y is an array of size Ny = size(y), then these conditions apply:

    • If xq is a scalar or vector, then size(p) returns [Ny(1:end-1) length(xq)].

    • If xq is an array, then size(p) returns [Ny(1:end-1) size(xq).

pp — Piecewise polynomial
structure

Piecewise polynomial, returned as a structure. Use this structurewith the ppval function toevaluate the interpolating polynomials at one or more query points.The structure has these fields.

FieldDescription
form

'pp' for piecewise polynomial

breaks

Vector of lengthL+1withstrictly increasing elements that represent the start and end of eachofLintervals

coefs

L-by-k matrix witheach rowcoefs(i,:)containing the localcoefficients of an orderkpolynomialon theith interval,[breaks(i),breaks(i+1)]

pieces

Number of pieces, L

order

Order of the polynomials

dim

Dimensionality of target

Since the polynomial coefficients in coefs arelocal coefficients for each interval, you must subtract the lowerendpoint of the corresponding knot interval to use the coefficientsin a conventional polynomial equation. In other words, for the coefficients [a,b,c,d] onthe interval [x1,x2], the corresponding polynomialis

f(x)=a(x−x1)3+b(x−x1)2+c(x−x1)+d .

More About

collapse all

Shape-Preserving Piecewise Cubic Interpolation

pchip interpolates usinga piecewise cubic polynomial P(x) withthese properties:

  • On each subinterval xk≤x≤xk+1,the polynomial P(x) is a cubic Hermiteinterpolating polynomial for the given data points with specifiedderivatives (slopes) at the interpolation points.

  • P(x) interpolates y,that is, P(xj)=yj, and the first derivative dPdx is continuous. The second derivative d2Pdx2 is probably not continuous sojumps at the xj are possible.

  • The cubic interpolant P(x) is shape preserving. The slopesat the xj are chosen in such a way that P(x) preserves the shape of the dataand respects monotonicity. Therefore, on intervals where the datais monotonic, so is P(x), and at pointswhere the data has a local extremum, so does P(x).

Note

If y is a matrix, P(x) satisfies these properties foreach row of y.

Tips

  • spline constructs S(x) in almost the same way pchip constructs P(x). However, spline choosesthe slopes at the xj differently, namelyto make even S″(x) continuous. This differencehas several effects:

    • spline produces a smoother result,such that S″(x) is continuous.

    • spline produces a more accurateresult if the data consists of values of a smooth function.

    • pchip has no overshoots and lessoscillation if the data is not smooth.

    • pchip is less expensive to setup.

    • The two are equally expensive to evaluate.

References

[1] Fritsch, F. N. and R. E. Carlson. "MonotonePiecewise Cubic Interpolation." SIAM Journal on NumericalAnalysis. Vol. 17, 1980, pp.238–246.

[2] Kahaner, David, Cleve Moler, Stephen Nash. NumericalMethods and Software. Upper Saddle River, NJ: PrenticeHall, 1988.

Extended Capabilities

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

See Also

interp1 | spline | makima | ppval

MATLAB Command

You clicked a link that corresponds to this MATLAB command: 

 
Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.
 
 
 
 
 
 
 
 Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (4) 
 
 
 
 
Select a Web Site
 
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
 
 
 
You can also select a web site from the following list:
 
 
 
Americas
 
     
  • América Latina (Español)
    •  
  • Canada (English)
    •  
  • United States (English)
    •  
 
 
 
Europe
 
 
 
     
  • Belgium (English)
    •  
  • Denmark (English)
    •  
  • Deutschland (Deutsch)
    •  
  • España (Español)
    •  
  • Finland (English)
    •  
  • France (Français)
    •  
  • Ireland (English)
    •  
  • Italia (Italiano)
    •  
  • Luxembourg (English)
    •  
 
 
 
     
  • Netherlands (English)
    •  
  • Norway (English)
    •  
  • Österreich (Deutsch)
    •  
  • Portugal (English)
    •  
  • Sweden (English)
    •  
  • Switzerland 
       
    • Deutsch
      •  
    • English
      •  
    • Français
      •  
     
    •  
  • United Kingdom (English)
    •  
 
 
 
 
 
Asia Pacific
  
 
 
Contact your local office
 
 
 
 
 

                                                        Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) (2024)
                        

                            
                        

                    

                                                                                                            
Top Articles

                            

                                                                    



    
    

        Creed 3 Cast & Characters: 14 Main Actors and Who They Play
    


                                                                    



    
    

        Creed III: The Cast List, Including Michael B. Jordan And Jonathan Majors
    


                                                                                                    



    
    

        Here Are the Net Worth and Income That Put You in the Top 5% of American Households by Age | The Motley Fool
    


                



    
    

        How YouTube became a TV powerhouse and how much ad revenue it makes
    


                



    
    

        All-Time Top Battlers (Most Battle Points)
    


                



    
    

        All-Time Top Ant Exterminators
    


                



    
    

        Austin’s Craigslist: Your Ultimate Guide to Buying, Selling, and Discovering
    


                



    
    

        Word Chum Cheats Grabber
    


                



    
    

        Craigslist Georgia Pets
    


                



    
    

        Miniature Dachshund Puppies For Sale California Craigslist
    


                                                                

                                                                
Latest Posts

                        

                                                            



    
    

        Erfelijke darmkanker - Incidentie en risico’s lynch - Richtlijn
    


                                                            



    
    

        Erfelijke darmkanker - Lynch syndroom colosco - Richtlijn
    


                                                    

                    
                    

        

    
    
    



                    

    
Article information

    
Author:  Gregorio Kreiger 

    
Last Updated:   

    
Views: 6299

    
Rating: 4.7 / 5 (57 voted)

    
Reviews: 88% of readers found this page helpful





    
Author information

    
Name: Gregorio Kreiger

    
Birthday: 1994-12-18

    
Address: 89212 Tracey Ramp, Sunside, MT 08453-0951

    
Phone: +9014805370218

    
Job: Customer Designer

    
Hobby: Mountain biking, Orienteering, Hiking, Sewing, Backpacking, Mushroom hunting, Backpacking

    
Introduction: My name is Gregorio Kreiger, I am a tender, brainy, enthusiastic, combative, agreeable, gentle, gentle person who loves writing and wants to share my knowledge and understanding with you.