Modified Akima piecewise cubic Hermite interpolation
Use makima
to interpolate a cosine curve over unevenly spaced sample points.
x = [0 1 2.5 3.6 5 7 8.1 10]; y = cos(x); xq = 0:.25:10; yq = makima(x,y,xq); plot(x,y,'o',xq,yq,'')
With oscillatory functions, the Akima algorithm flattens the curve near local extrema. To compensate for this flattening, you can add more sample points near the local extrema.
Add sample points near $\mathit{x}=6.5$ and $\mathit{x}=9$ and replot the interpolation.
x = [0 1 2.5 3.6 5 6.5 7 8.1 9 10]; y = cos(x); xq = 0:.25:10; yq = makima(x,y,xq); plot(x,y,'o',xq,yq,'')
spline
, pchip
, and makima
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')
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.
Create vectors for the sample points x
and values at those points y
. Use makima
to construct a piecewise polynomial structure for the data.
x = 5:5; y = [1 1 1 0 0 1 1 2 2 2 2]; pp = makima(x,y)
pp = struct with fields:
form: 'pp'
breaks: [5 4 3 2 1 0 1 2 3 4 5]
coefs: [10x4 double]
pieces: 10
order: 4
dim: 1
The structure contains the information for 10 polynomials of order 4 that span the data. pp.coefs(i,:)
contains the coefficients for the polynomial that is valid in the region defined by the breakpoints [breaks(i) breaks(i+1)]
.
Use the structure with ppval
to evaluate the interpolation at several query points, and then plot the results. In regions with three or more constant points, the Akima algorithm connects the points with a straight line.
xq = 5:0.2:5; m = ppval(pp,xq); plot(x,y,'o',xq,m,'.') ylim([0.2 2.2])
x
— Sample pointsSample points, specified as a vector. The vector x
specifies the
points at which the data y
is given. The elements of
x
must be unique.
Data Types: single
 double
y
— Function values at sample pointsFunction values at sample points, specified as a numeric vector, matrix, or array.
x
and y
must have the same length.
If y
is a matrix or array, then the values in the last dimension,
y(:,...,:,j)
, are taken as the values to match with
x
. In that case, the last dimension of y
must be
the same length as x
.
Data Types: single
 double
xq
— Query pointsQuery points, specified as a scalar, vector, matrix, or array. The points specified
in xq
are the xcoordinates for the interpolated
function values yq
computed by makima
.
Data Types: single
 double
yq
— Interpolated values at query pointsInterpolated values at query points, returned as a scalar, vector, matrix, or array.
The size of yq
is related to the sizes of y
and
xq
:
If y
is a vector, then yq
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(yq)
returns [Ny(1:end1)
length(xq)]
.
If xq
is an array, then size(yq)
returns [Ny(1:end1) size(xq)]
.
pp
— Piecewise polynomialPiecewise polynomial, returned as a structure. Use this structure with the ppval
function to evaluate the interpolating polynomials at one or more
query points. The structure has these fields.
Field  Description 

form 

breaks  Vector of length 
coefs 

pieces  Number of pieces, 
order  Order of polynomials 
dim  Dimensionality of target 
Because the polynomial coefficients in coefs
are local
coefficients for each interval, you must subtract the lower endpoint of the
corresponding knot interval to use the coefficients in a conventional polynomial
equation. In other words, for the coefficients [a,b,c,d]
on the
interval [x1,x2]
, the corresponding polynomial is
$$f\left(x\right)=a{\left(x{x}_{1}\right)}^{3}+b{\left(x{x}_{1}\right)}^{2}+c\left(x{x}_{1}\right)+d\text{\hspace{0.17em}}.$$
The Akima algorithm for onedimensional interpolation, described in [1] and [2], performs cubic interpolation to produce piecewise polynomials with continuous firstorder derivatives (C1). The algorithm avoids excessive local undulations.
If $${\delta}_{i}=\frac{{v}_{i+1}{v}_{i}}{{x}_{i+1}{x}_{i}}$$ is the slope on interval $$\left[{x}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{i+1}\right)$$, then the value of the derivative $${d}_{i}$$ at the sample point $${x}_{i}$$ is a weighted average of nearby slopes:
$${d}_{i}=\frac{{w}_{1}}{{w}_{1}+{w}_{2}}{\delta}_{i1}+\frac{{w}_{2}}{{w}_{1}+{w}_{2}}{\delta}_{i}.$$
In Akima's original formula, the weights are:
$$\begin{array}{l}{w}_{1}=\left{\delta}_{i+1}{\delta}_{i}\right,\\ {w}_{2}=\left{\delta}_{i1}{\delta}_{i2}\right.\end{array}$$
The original Akima algorithm gives equal weight to the points on both sides, evenly dividing an undulation.
When two flat regions with different slopes meet, the modification made to the original Akima algorithm gives more weight to the side where the slope is closer to zero. This modification gives priority to the side that is closer to horizontal, which is more intuitive and avoids overshoot. In particular, whenever there are three or more consecutive collinear points, the algorithm connects them with a straight line and thus avoids an overshoot.
The weights used in the modified Akima algorithm are:
$$\begin{array}{l}{w}_{1}=\left{\delta}_{i+1}{\delta}_{i}\right+\frac{\left{\delta}_{i+1}+{\delta}_{i}\right}{2},\\ {w}_{2}=\left{\delta}_{i1}{\delta}_{i2}\right+\frac{\left{\delta}_{i1}+{\delta}_{i2}\right}{2}.\end{array}$$
Compared to the spline
algorithm, the Akima algorithm produces
fewer undulations and is better suited to deal with quick changes between flat regions.
Compared to the pchip
algorithm, the Akima algorithm is not as
aggressively flattened and is therefore still able to deal with oscillatory data.
[1] Akima, Hiroshi. "A new method of interpolation and smooth curve fitting based on local procedures." Journal of the ACM (JACM) , 17.4, 1970, pp. 589–602.
[2] Akima, Hiroshi. "A method of bivariate interpolation and smooth surface fitting based on local procedures." Communications of the ACM , 17.1, 1974, pp. 18–20.
A modified version of this example exists on your system. Do you want to open this version instead?
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.
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: .
Select web siteYou can also select a web site from the following list:
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.