UNIT 2 • EQUAL INTERVAL INTERPOLATION

Finite Difference Table
Forward, Backward & Central

A finite difference table lists the values y and their differences: $\Delta y_i = y_{i+1} - y_i$ , then the differences of those, and so on. It is the starting point of Newton's, Gauss's, Stirling's and Bessel's formulas.

THE OPERATORSEQUAL SPACING h

\begin{gathered} \Delta y_i = y_{i+1} - y_i, \quad \nabla y_i = y_i - y_{i-1}, \quad \delta y_{i + \frac{1}{2}} = y_{i+1} - y_i \\ E = 1 + \Delta, \quad \nabla = 1 - E^{-1}, \quad \delta = E^{1/2} - E^{-1/2} \end{gathered}

1. Check the SpacingThe x values must go up in equal steps h.

2. Subtract NeighboursEach column is the difference of the one before: lower minus upper.

3. Stop When ConstantWhen a column is constant (the next is all 0), you have found the degree.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Make the forward difference table for x = 1, 2, 3, 4, 5 and y = 1, 8, 27, 64, 125, and find the degree of the polynomial.

STEP 01 • FIRST DIFFERENCES

\Delta y: \ 8 - 1 = 7, \quad 27 - 8 = 19, \quad 64 - 27 = 37, \quad 125 - 64 = 61

STEP 02 • HIGHER DIFFERENCES

\Delta^2 y: \ 12, \ 18, \ 24 \qquad \Delta^3 y: \ 6, \ 6 \qquad \Delta^4 y: \ 0

The third differences are constant and the fourth is 0.

FINAL CONCLUSION

Constant third differences: a cubic

\Delta^3 y = 6 \implies \text{degree } 3 \ (\text{here } y = x^3)

The calculator below opens on this data. Switch the notation to see the same numbers labelled ∇ or δ.

LABORATORY

Interactive Difference Table

LABORATORY

Difference Table Engine

x values (equally spaced)y values

Notation

Common Questions

What is the difference between Δ, ∇ and δ?

They are the same subtractions, labelled from different ends. Forward:

\Delta y_i = y_{i+1} - y_i

. Backward:

\nabla y_i = y_i - y_{i-1}

. Central:

\delta y_{i+1/2} = y_{i+1} - y_i

. So

\Delta y_0 = \nabla y_1 = \delta y_{1/2}

What is the shift operator E?

E\,y_i = y_{i+1}

. The others follow from it:

\Delta = E - 1

\nabla = 1 - E^{-1}

and

\delta = E^{1/2} - E^{-1/2}

Why do the differences become constant?

For a polynomial of degree n with equally spaced x, the nth differences are constant and the higher ones are 0. So the table tells you the lowest degree of polynomial that fits the data.

PreviousGauss Backward Formula

Unit 2•12 of 63

NextStirling's Formula

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 2 • EQUAL INTERVAL INTERPOLATION

Finite Difference Table
Forward, Backward & Central

THE OPERATORSEQUAL SPACING h

\begin{gathered} \Delta y_i = y_{i+1} - y_i, \quad \nabla y_i = y_i - y_{i-1}, \quad \delta y_{i + \frac{1}{2}} = y_{i+1} - y_i \\ E = 1 + \Delta, \quad \nabla = 1 - E^{-1}, \quad \delta = E^{1/2} - E^{-1/2} \end{gathered}

1. Check the SpacingThe x values must go up in equal steps h.

2. Subtract NeighboursEach column is the difference of the one before: lower minus upper.

3. Stop When ConstantWhen a column is constant (the next is all 0), you have found the degree.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Make the forward difference table for x = 1, 2, 3, 4, 5 and y = 1, 8, 27, 64, 125, and find the degree of the polynomial.

STEP 01 • FIRST DIFFERENCES

\Delta y: \ 8 - 1 = 7, \quad 27 - 8 = 19, \quad 64 - 27 = 37, \quad 125 - 64 = 61

STEP 02 • HIGHER DIFFERENCES

\Delta^2 y: \ 12, \ 18, \ 24 \qquad \Delta^3 y: \ 6, \ 6 \qquad \Delta^4 y: \ 0

The third differences are constant and the fourth is 0.

FINAL CONCLUSION

Constant third differences: a cubic

\Delta^3 y = 6 \implies \text{degree } 3 \ (\text{here } y = x^3)

The calculator below opens on this data. Switch the notation to see the same numbers labelled ∇ or δ.

LABORATORY

Interactive Difference Table

LABORATORY

Difference Table Engine

x values (equally spaced)y values

Notation

Common Questions

What is the difference between Δ, ∇ and δ?

They are the same subtractions, labelled from different ends. Forward:

\Delta y_i = y_{i+1} - y_i

. Backward:

\nabla y_i = y_i - y_{i-1}

. Central:

\delta y_{i+1/2} = y_{i+1} - y_i

. So

\Delta y_0 = \nabla y_1 = \delta y_{1/2}

What is the shift operator E?

E\,y_i = y_{i+1}

. The others follow from it:

\Delta = E - 1

\nabla = 1 - E^{-1}

and

\delta = E^{1/2} - E^{-1/2}

Why do the differences become constant?

For a polynomial of degree n with equally spaced x, the nth differences are constant and the higher ones are 0. So the table tells you the lowest degree of polynomial that fits the data.

PreviousGauss Backward Formula

Unit 2•12 of 63

NextStirling's Formula

Finite Difference Table Forward, Backward & Central

Step-by-Step Worked Example

Constant third differences: a cubic

Interactive Difference Table

Difference Table Engine

Common Questions

NETZ Engine

Finite Difference Table Forward, Backward & Central

Step-by-Step Worked Example

Constant third differences: a cubic

Interactive Difference Table

Difference Table Engine

Common Questions

Finite Difference Table
Forward, Backward & Central

Finite Difference Table
Forward, Backward & Central