UNIT 2 • EQUAL INTERVAL INTERPOLATION

Bessel's Formula
Interpolating Halfway

Bessel's formula interpolates between two middle points $x_0$ and $x_1$ of an equally spaced table. It starts from their average, $\frac{y_0 + y_1}{2}$ , and adds central-difference corrections.

BESSEL'S FORMULABEST FOR ¼ ≤ p ≤ ¾

\begin{gathered} y = \frac{y_0 + y_1}{2} + \left(p - \tfrac{1}{2}\right)\Delta y_0 \\ + \frac{p(p - 1)}{2!}\cdot\frac{\Delta^2 y_{-1} + \Delta^2 y_0}{2} \\ + \frac{\left(p - \tfrac{1}{2}\right)p(p - 1)}{3!}\Delta^3 y_{-1} + \cdots \\ p = \frac{x - x_0}{h} \end{gathered}

1. Pick x₀ and x₁The two tabulated points either side of x.

2. Find p

p = (x - x_0)/h

, somewhere between 0 and 1, ideally near ½.

3. Average, then CorrectStart from the mean of y₀ and y₁; even orders use the mean of two differences.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

From x = 1, 2, 3, 4, 5 and y = 1, 8, 27, 64, 125, estimate y at x = 3.5.

STEP 01 • CHOOSE x₀ AND p

x_0 = 3, \ x_1 = 4, \quad p = \frac{3.5 - 3}{1} = 0.5

At $p = \tfrac{1}{2}$ the factor $p - \tfrac{1}{2} = 0$ , so every odd-order term drops out.

STEP 02 • THE TERMS THAT REMAIN

\frac{y_0 + y_1}{2} = \frac{27 + 64}{2} = 45.5

\frac{p(p - 1)}{2}\cdot\frac{\Delta^2 y_{-1} + \Delta^2 y_0}{2} = \frac{0.5 \times (-0.5)}{2} \times \frac{18 + 24}{2} = -2.625

STEP 03 • ADD THEM UP

y(3.5) \approx 45.5 - 2.625 = 42.875

The data come from $y = x^3$ , and $3.5^3 = 42.875$ exactly.

FINAL CONCLUSION

y(3.5) ≈ 42.875

y(3.5) = 42.875

The calculator below opens on this table and tints the differences it used.

LABORATORY

Interactive Bessel Calculator

LABORATORY

Bessel Engine

x values (equally spaced)y values

Find y at x =

Common Questions

When should you use Bessel's formula?

When x is roughly halfway between two middle points of the table: it works best for

\tfrac{1}{4} \le p \le \tfrac{3}{4}

. At exactly

p = \tfrac{1}{2}

every odd-order term is 0, which makes the working short.

What is the difference between Bessel's and Stirling's formulas?

Stirling centres on one tabulated point and suits p near 0. Bessel centres on the midpoint between two points and suits p near ½. Both use central differences.

Why does the formula start with the mean of y₀ and y₁?

Because it is built around the midpoint between

x_0

and

x_1

. The later terms correct that first guess using the curvature of the data.

PreviousStirling's Formula

Unit 2•14 of 63

NextNewton's Divided Difference Formula

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 2 • EQUAL INTERVAL INTERPOLATION

Bessel's Formula
Interpolating Halfway

Bessel's formula interpolates between two middle points $x_0$ and $x_1$ of an equally spaced table. It starts from their average, $\frac{y_0 + y_1}{2}$ , and adds central-difference corrections.

BESSEL'S FORMULABEST FOR ¼ ≤ p ≤ ¾

\begin{gathered} y = \frac{y_0 + y_1}{2} + \left(p - \tfrac{1}{2}\right)\Delta y_0 \\ + \frac{p(p - 1)}{2!}\cdot\frac{\Delta^2 y_{-1} + \Delta^2 y_0}{2} \\ + \frac{\left(p - \tfrac{1}{2}\right)p(p - 1)}{3!}\Delta^3 y_{-1} + \cdots \\ p = \frac{x - x_0}{h} \end{gathered}

1. Pick x₀ and x₁The two tabulated points either side of x.

2. Find p

p = (x - x_0)/h

, somewhere between 0 and 1, ideally near ½.

3. Average, then CorrectStart from the mean of y₀ and y₁; even orders use the mean of two differences.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

From x = 1, 2, 3, 4, 5 and y = 1, 8, 27, 64, 125, estimate y at x = 3.5.

STEP 01 • CHOOSE x₀ AND p

x_0 = 3, \ x_1 = 4, \quad p = \frac{3.5 - 3}{1} = 0.5

At $p = \tfrac{1}{2}$ the factor $p - \tfrac{1}{2} = 0$ , so every odd-order term drops out.

STEP 02 • THE TERMS THAT REMAIN

\frac{y_0 + y_1}{2} = \frac{27 + 64}{2} = 45.5

\frac{p(p - 1)}{2}\cdot\frac{\Delta^2 y_{-1} + \Delta^2 y_0}{2} = \frac{0.5 \times (-0.5)}{2} \times \frac{18 + 24}{2} = -2.625

STEP 03 • ADD THEM UP

y(3.5) \approx 45.5 - 2.625 = 42.875

The data come from $y = x^3$ , and $3.5^3 = 42.875$ exactly.

FINAL CONCLUSION

y(3.5) ≈ 42.875

y(3.5) = 42.875

The calculator below opens on this table and tints the differences it used.

LABORATORY

Interactive Bessel Calculator

LABORATORY

Bessel Engine

x values (equally spaced)y values

Find y at x =

Common Questions

When should you use Bessel's formula?

When x is roughly halfway between two middle points of the table: it works best for

\tfrac{1}{4} \le p \le \tfrac{3}{4}

. At exactly

p = \tfrac{1}{2}

every odd-order term is 0, which makes the working short.

What is the difference between Bessel's and Stirling's formulas?

Stirling centres on one tabulated point and suits p near 0. Bessel centres on the midpoint between two points and suits p near ½. Both use central differences.

Why does the formula start with the mean of y₀ and y₁?

Because it is built around the midpoint between

x_0

and

x_1

. The later terms correct that first guess using the curvature of the data.

PreviousStirling's Formula

Unit 2•14 of 63

NextNewton's Divided Difference Formula

Bessel's Formula Interpolating Halfway

Step-by-Step Worked Example

y(3.5) ≈ 42.875

Interactive Bessel Calculator

Bessel Engine

Common Questions

NETZ Engine

Bessel's Formula Interpolating Halfway

Step-by-Step Worked Example

y(3.5) ≈ 42.875

Interactive Bessel Calculator

Bessel Engine

Common Questions

Bessel's Formula
Interpolating Halfway

Bessel's Formula
Interpolating Halfway