Modified Akima piecewise cubic Hermite interpolation
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 and 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,'--')
Compare the interpolation results produced by
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
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,
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,
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
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
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 points
Sample 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.
y— Function values at sample points
Function values at sample points, specified as a numeric vector, matrix, or array.
y must have the same length.
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
xq— Query points
Query points, specified as a scalar, vector, matrix, or array. The points specified
xq are the x-coordinates for the interpolated
yq computed by
yq— Interpolated values at query points
Interpolated values at query points, returned as a scalar, vector, matrix, or array.
The size of
yq is related to the sizes of
y is a vector, then
yq has the same
y is an array of size
Ny = size(y),
then these conditions apply:
xq is a scalar or vector, then
xq is an array, then
pp— Piecewise polynomial
Piecewise 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.
Vector of length
Number of pieces,
Order of polynomials
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
[x1,x2], the corresponding polynomial is
The Akima algorithm for one-dimensional interpolation, described in  and , performs cubic interpolation to produce piecewise polynomials with continuous first-order derivatives (C1). The algorithm avoids excessive local undulations.
If is the slope on interval , then the value of the derivative at the sample point is a weighted average of nearby slopes:
In Akima's original formula, the weights are:
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:
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.
 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.
 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.