MATLAB Function Reference |

One-dimensional data interpolation (table lookup)

**Syntax**

yi = interp1(x,Y,xi) yi = interp1(Y,xi) yi = interp1(x,Y,xi,method) yi = interp1(x,Y,xi,method,'extrap') yi = interp1(x,Y,xi,method,extrapval) pp = interp1(x,Y,method,'pp')

**Description**

`yi = interp1(x,Y,xi)`

interpolates to find `yi`

, the values of the underlying function `Y`

at the points in the vector or array `xi`

. `x`

must be a vector. Y can be a scalar, a vector, or an array of any dimension, subject to the following conditions:

- If
`Y`

is a scalar or vector, it must have the same length as`x`

. A scalar value for`x`

or`Y`

is expanded to have the same length as the other.`xi`

can be a scalar, a vector, or a multidimensional array, and`yi`

has the same size as`xi`

. - If
`Y`

is an array that is not a vector, the size of`Y`

must have the form`[n,d1,d2,...,dk]`

, where`n`

is the length of`x`

. The interpolation is performed for each`d1`

-by-`d2`

-by-...-`dk`

value in`Y`

. The sizes of`xi`

and`yi`

are related as follows:

```
yi = interp1(Y,xi)
```

assumes that `x = 1:N`

, where `N`

is the length of `Y`

for vector `Y`

, or `size(Y,1)`

for matrix `Y`

.

interpolates using alternative methods:`yi = interp1(x,Y,xi,`

`method`

```
)
```

For the `'nearest'`

, `'linear'`

, and `'v5cubic'`

methods, `interp1(x,Y,xi,method)`

returns `NaN`

for any element of `xi`

that is outside the interval spanned by `x`

. For all other methods, `interp1`

performs extrapolation for out of range values.

```
yi = interp1(x,Y,xi,method,'extrap')
```

uses the specified method to perform extrapolation for out of range values.

```
yi = interp1(x,Y,xi,method,extrapval)
```

returns the scalar `extrapval`

for out of range values. `NaN`

and `0`

are often used for `extrapval`

.

`pp = interp1(x,Y,method,'pp')`

uses the specified method to generate the piecewise polynomial form (ppform) of `Y`

. You can use any of the methods in the preceding table, except for `'v5cubic'`

.

The `interp1`

command interpolates between data points. It finds values at intermediate points, of a one-dimensional function that underlies the data. This function is shown below, along with the relationship between vectors `x`

, `Y`

, `xi`

, and `yi`

.

Interpolation is the same operation as *table lookup*. Described in table lookup terms, the *table* is `[x,Y]`

and `interp1`

*looks up* the elements of `xi`

in `x`

, and, based upon their locations, returns values `yi`

interpolated within the elements of `Y`

.

**Examples**

**Example 1.** Generate a coarse sine curve and interpolate over a finer abscissa.

**Example 2.** The following multidimensional example creates 2-by-2 matrices of interpolated function values, one matrix for each of the three functions *x*^{2}, *x*^{3}, and *x*^{4}.

The result `yi`

has size 2-by-2-by-3.

**Example 3.** Here are two vectors representing the census years from 1900 to 1990 and the corresponding United States population in millions of people.

t = 1900:10:1990; p = [75.995 91.972 105.711 123.203 131.669... 150.697 179.323 203.212 226.505 249.633];

The expression `interp1(t,p,1975)`

interpolates within the census data to estimate the population in 1975. The result is

Now interpolate within the data at every year from 1900 to 2000, and plot the result.

Sometimes it is more convenient to think of interpolation in table lookup terms, where the data are stored in a single table. If a portion of the census data is stored in a single 5-by-2 table,

then the population in 1975, obtained by table lookup within the matrix `tab`

, is

**Example 4.** The following example uses the `'cubic'`

method to generate the piecewise polynomial form (ppform) of `Y`

, and then evaluates the result using `ppval`

.

x = 0:.2:pi; y = sin(x); pp = interp1(x,y,'cubic','pp'); xi = 0:.1:pi; yi = ppval(pp,xi); plot(x,y,'ko'), hold on, plot(xi,yi,'r:'), hold off

**Algorithm**

The `interp1`

command is a MATLAB M-file. The '`nearest'`

and '`linear'`

methods have straightforward implementations.

For the '`spline'`

method, `interp1`

calls a function `spline`

that uses the functions `ppval`

, `mkpp`

, and `unmkpp`

. These routines form a small suite of functions for working with piecewise polynomials. `spline`

uses them to perform the cubic spline interpolation. For access to more advanced features, see the `spline`

reference page, the M-file help for these functions, and the Spline Toolbox.

For the '`pchip'`

and `'cubic'`

methods, `interp1`

calls a function `pchip`

that performs piecewise cubic interpolation within the vectors `x`

and `y`

. This method preserves monotonicity and the shape of the data. See the `pchip`

reference page for more information.

**See Also**

`interpft`

, `interp2`

, `interp3`

, `interpn`

, `pchip`

, `spline`

**References**

[1] de Boor, C., *A Practical Guide to Splines*, Springer-Verlag, 1978.

int8, int16, int32, int64 | interp2 |

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