# INTERPOLATE - Interpolation and Extrapolation

estimate the value of the function represented by (x,y) data set at an intermediate x-value.

## Syntax

INTERPOLATE(X, Y, target, Method, extrapolate)

X is the x-component of the input data table (a one dimensional array of cells (e.g. rows or columns)).

Y is the y-component (i.e. function) of the input data table (a one dimensional array of cells (e.g. rows or columns)).

target is the desired x-value(s) to interpolate for (a single value or a one dimensional array of cells (e.g. rows or columns)).

Method is the interpolation method (1=Forward Flat, 2=Backward Flat, 3=Linear, 4=Cubic Spline).

Value Method
1 Forward Flat
2 Backward Flat
3 Linear (default)
4 Cubic Spline

extrapolate sets whether or not to allow extrapolation (1=Yes, 0=No). If missing, the default is to not allow extrapolation.

Value Extrapolate
0 No (default)
1 Yes

## Remarks

1. The X and Y array sizes must be identical.
2. The X-array and Y-array both consist of numerical values. Dates in Excel are internally represented by numbers.
3. The values in the X-array can be unsorted and may have duplicate values.
4. In the case where X has duplicate values, INTERPOLATE will replace those duplicate values with a single entry, setting the corresponding y-value equal to the average.
5. The X and/or Y arrays may have missing values (#N/A). In this case, INTERPOLATE will remove those entries.
6. For cubic spline interpolation, we construct a set of natural cubic splines that are twice continuously differentiable functions to yield the least oscillation about the function f which is interpolated.

## Examples

Example 1: *Interpolation (ordered X and no missing values)

 1 2 3 4 5 6 7 8 9 10 11
A B
X Y
0.10 0.01
0.50 0.25
1.00 1.00
1.50 2.25
2.00 4.00
2.20 4.84
2.60 6.76
2.80 7.84
3.00 9.00
4.00 16.00

Formula Description (Result)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,1,0) Forward-flat interpolation (4)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,2,0) Backward-flat interpolation (4.84)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,3,0) Linear interpolation (4.63)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,4,0) Cubic spline interpolation (4.619)

Example 2: *Extrapolation (ordered X and no missing values)

 1 2 3 4 5 6 7 8 9 10 11
A B
X Y
0.10 0.01
0.50 0.25
1.00 1.00
1.50 2.25
2.00 4.00
2.20 4.84
2.60 6.76
2.80 7.84
3.00 9.00
4.00 16.00

Formula Description (Result)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,1,1) Forward-flat interpolation (4)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,2,1) Backward-flat interpolation (4.84)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,3,1) Linear interpolation (4.63)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,4,1) Cubic spline interpolation (4.619)

Example 3: *Interpolation (un-ordered X and no missing values)

 1 2 3 4 5 6 7 8 9 10 11
A B
X Y
2 4.00
1.00 1.00
0.50 0.25
1.50 2.25
0.10 0.01
2.20 4.84
2.60 6.76
2.80 7.84
1.00 1.00
3.00 9.00

Formula Description (Result)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,1,0) Forward-flat interpolation (4)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,2,0) Backward-flat interpolation (4.84)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,3,0) Linear interpolation (4.63)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,4,0) Cubic spline interpolation (4.619)

Example 4: *Interpolation (un-ordered X, no duplicates, and with missing values)

 1 2 3 4 5 6 7 8 9 10 11
A B
X Y
2 4.00
1.00 1.00
0.50 0.25
1.50 #N/A
0.10 0.01
2.20 4.48
2.60 6.76
2.80 7.84
1.00 #N/A
3.00 9.00

Formula Description (Result)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,1,0) Forward-flat interpolation (4)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,2,0) Backward-flat interpolation (4.48)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,3,0) Linear interpolation (4.36)
=INTERPOLATE($A$2:$A$11,$B$2:$B$11,2.15,4,0) Cubic spline interpolation (4.335)

Example 5: *Interpolation (un-ordered X, with duplicates, and no missing values)

 1 2 3 4 5 6 7 8 9 10 11 12 13
A B
X Y
2 4.00
1.00 1.05
1.00 1.00
0.50 0.25
1.50 2.25
1.50 2.10
0.10 0.01
2.20 4.84
2.60 6.76
2.80 7.84
1.00 1.30
3.00 9.00

Formula Description (Result)
=INTERPOLATE($A$2:$A$13,$B$2:$B$13,2.15,1,0) Forward-flat interpolation (4)
=INTERPOLATE($A$2:$A$13,$B$2:$B$13,2.15,2,0) Backward-flat interpolation (4.84)
=INTERPOLATE($A$2:$A$13,$B$2:$B$13,2.15,3,0) Linear interpolation (4.63)
=INTERPOLATE($A$2:$A$13,$B$2:$B$13,2.15,4,0) Cubic spline interpolation (4.619)

## References

• Kincaid, David; Ward Cheney (2002). Numerical Analysis (3rd edition). Brooks/Cole. ISBN 0-534-38905-8. Chapter 6.
• Ahlberg, Nielson, and Walsh, The Theory of Splines and Their Applications, 1967.