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# pchip

Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)

## Syntax

``p = pchip(x,y,xq)``
``pp = pchip(x,y)``

## Description

example

````p = pchip(x,y,xq)` returns a vector of interpolated values `p` corresponding to the query points in `xq`. The values of `p` are determined by shape-preserving piecewise cubic interpolation of `x` and `y`.```

example

````pp = pchip(x,y)` returns a piecewise polynomial structure for use with `ppval` and the spline utility `unmkpp`.```

## Examples

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Compare the interpolation results produced by `spline` and `pchip` for two different functions.

Create vectors of `x` values, function values at those points `y`, and query points `xq`. Compute interpolations at the query points using both `spline` and `pchip`. 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); plot(x,y,'o',xq1,p,'-',xq1,s,'-.') legend('Sample Points','pchip','spline','Location','SouthEast')```

In this case, `pchip` is favorable since it does not oscillate as freely between the sample points.

Perform a second comparison using an oscillatory sample function.

```x = 0:25; y = besselj(1,x); xq2 = 0:0.01:25; p = pchip(x,y,xq2); s = spline(x,y,xq2); plot(x,y,'o',xq2,p,'-',xq2,s,'-.') legend('Sample Points','pchip','spline')```

When the underlying function is oscillatory, `spline` captures the movement between points better than `pchip`.

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])```

## Input Arguments

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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.

Data Types: `single` | `double`

Function 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`

Query points, specified as a vector. The points specified in `xq` are the x-coordinates for the interpolated function values `p` computed by `pchip`.

Data Types: `single` | `double`

## Output Arguments

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Interpolated values at query points, returned as a vector, matrix, or array. The size of `p` depends on the sizes of the inputs:

• If `y` is a vector, then `p` is a vector that has the same length as `xq`.

• If `y` has two or more dimensions denoted by `n`, then `p` has size ```[size(y,1) size(y,2) ... size(y,n-1) length(xq)]```. For example, if `y` is a matrix, then `p` is of size ```[size(y,1) length(xq)]```.

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.

FieldDescription
`form`

`'pp'` for piecewise polynomial

`breaks`

Vector of length `L+1` with strictly increasing elements that represent the start and end of each of` L` intervals

`coefs`

`L`-by-`k` matrix with each row `coefs(i,:)` containing the local coefficients of an order `k` polynomial on the `i`th 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` 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

`$f\left(x\right)=a{\left(x-{x}_{1}\right)}^{3}+b{\left(x-{x}_{1}\right)}^{2}+c\left(x-{x}_{1}\right)+d\text{\hspace{0.17em}}.$`

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### Shape-Preserving Piecewise Cubic Interpolation

`pchip` interpolates using a piecewise cubic polynomial $P\left(x\right)$ with these properties:

• On each subinterval ${x}_{k}\le x\le {x}_{k+1}$, the polynomial $P\left(x\right)$ is a cubic Hermite interpolating polynomial for the given data points with specified derivatives (slopes) at the interpolation points.

• $P\left(x\right)$ interpolates y, that is, $P\left({x}_{j}\right)={y}_{j}$, and the first derivative $\frac{dP}{dx}$ is continuous. The second derivative $\frac{{d}^{2}P}{d{x}^{2}}$ is probably not continuous so jumps at the ${x}_{j}$ are possible.

• The cubic interpolant $P\left(x\right)$ is shape preserving. The slopes at the ${x}_{j}$ are chosen in such a way that $P\left(x\right)$ preserves the shape of the data and respects monotonicity. Therefore, on intervals where the data is monotonic, so is $P\left(x\right)$, and at points where the data has a local extremum, so does $P\left(x\right)$.

### Note

If y is a matrix, $P\left(x\right)$ satisfies these properties for each row of y.

## Tips

• `spline` constructs $S\left(x\right)$ in almost the same way `pchip` constructs $P\left(x\right)$. However, `spline` chooses the slopes at the ${x}_{j}$ differently, namely to make even ${S}^{″}\left(x\right)$ continuous. This difference has several effects:

• `spline` produces a smoother result, such that ${S}^{″}\left(x\right)$ is continuous.

• `spline` produces a more accurate result if the data consists of values of a smooth function.

• `pchip` has no overshoots and less oscillation if the data is not smooth.

• `pchip` is less expensive to set up.

• The two are equally expensive to evaluate.

## References

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

[2] Kahaner, David, Cleve Moler, Stephen Nash. Numerical Methods and Software. Upper Saddle River, NJ: Prentice Hall, 1988.