UNIT 2 • EQUAL INTERVAL INTERPOLATION

Gauss Forward
Central Difference Method

Gauss Forward Interpolation estimates functional values near the middle of an equally spaced table by taking forward zig-zag central difference steps around a central origin point $x_0$ .

GAUSS FORWARD FORMULAMID-RANGE ACCURACY

P(x) = y_0 + p \cdot \Delta y_0 + \frac{p(p-1)}{2!} \cdot \Delta^2 y_{-1} + \frac{(p+1) p (p-1)}{3!} \cdot \Delta^3 y_{-1} + \cdots

1. Central Parameter (p)

p = \frac{x - x_0}{h}

(distance from central origin

x_0

2. Equal Mesh Step (h)

h = x_{i+1} - x_i

(constant grid spacing).

3. Forward Central PathUses

\Delta y_0, \Delta^2 y_{-1}, \Delta^3 y_{-1}, \dots

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the functional dataset:

x	2.5	3.0	3.5	4.0	4.5
y	24.145	22.043	20.225	18.644	17.262

Let's estimate the value at $x = 3.75$ !

STEP 01 • SELECT CENTRAL ORIGIN & CALCULATE p

Select $x_0 = 3.5$ as central origin (closest point preceding $3.75$ ) with step size $h = 0.5$ :

p = \frac{x - x_0}{h} = \frac{3.75 - 3.5}{0.5} = \frac{0.25}{0.5} = 0.5

STEP 02 • CONSTRUCT CENTRAL DIFFERENCE TABLE

x	p	y	Δy	Δ²y	Δ³y
2.5	-2	24.145	-	-	-
3.0	-1	22.043	-1.818	-	-
3.5 (x_0)	0	20.225 (y_0)	-1.581 (Δy_0)	0.237 (Δ²y_-1)	-0.038 (Δ³y_-1)
4.0	1	18.644	-1.382	0.199	-
4.5	2	17.262	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Gauss Forward Polynomial Formula:

P(x) = y_0 + p \Delta y_0 + \frac{p(p-1)}{2!} \Delta^2 y_{-1} + \frac{(p+1)p(p-1)}{3!} \Delta^3 y_{-1}

Substituting values ( $p = 0.5, y_0 = 20.225, \Delta y_0 = -1.581, \Delta^2 y_{-1} = 0.237, \Delta^3 y_{-1} = -0.038$ ):

P(3.75) = 20.225 + (0.5)(-1.581) + \frac{(0.5)(-0.5)}{2}(0.237) + \frac{(1.5)(0.5)(-0.5)}{6}(-0.038)

P(3.75) = 20.22500 - 0.79050 - 0.02963 + 0.00238

P(3.75) = 19.40725

FINAL RESULT

Interpolated Value at x = 3.75: P(3.75) = 19.4072

\boxed{P(3.75) = 19.4072}

Using Gauss Forward central difference interpolation, the calculated value at $x = 3.75$ is exactly 19.4072!

LABORATORY

Interactive Live Calculator & Visualizer

CENTRAL DIFFERENCE INTERPOLATOR

Gauss Forward Calculation Engine

PreviousNewton Backward Interpolation

Unit 2•8 of 28

NextGauss Backward Interpolation

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 2 • EQUAL INTERVAL INTERPOLATION

Gauss Forward
Central Difference Method

Gauss Forward Interpolation estimates functional values near the middle of an equally spaced table by taking forward zig-zag central difference steps around a central origin point $x_0$ .

GAUSS FORWARD FORMULAMID-RANGE ACCURACY

P(x) = y_0 + p \cdot \Delta y_0 + \frac{p(p-1)}{2!} \cdot \Delta^2 y_{-1} + \frac{(p+1) p (p-1)}{3!} \cdot \Delta^3 y_{-1} + \cdots

1. Central Parameter (p)

p = \frac{x - x_0}{h}

(distance from central origin

x_0

2. Equal Mesh Step (h)

h = x_{i+1} - x_i

(constant grid spacing).

3. Forward Central PathUses

\Delta y_0, \Delta^2 y_{-1}, \Delta^3 y_{-1}, \dots

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the functional dataset:

x	2.5	3.0	3.5	4.0	4.5
y	24.145	22.043	20.225	18.644	17.262

Let's estimate the value at $x = 3.75$ !

STEP 01 • SELECT CENTRAL ORIGIN & CALCULATE p

Select $x_0 = 3.5$ as central origin (closest point preceding $3.75$ ) with step size $h = 0.5$ :

p = \frac{x - x_0}{h} = \frac{3.75 - 3.5}{0.5} = \frac{0.25}{0.5} = 0.5

STEP 02 • CONSTRUCT CENTRAL DIFFERENCE TABLE

x	p	y	Δy	Δ²y	Δ³y
2.5	-2	24.145	-	-	-
3.0	-1	22.043	-1.818	-	-
3.5 (x_0)	0	20.225 (y_0)	-1.581 (Δy_0)	0.237 (Δ²y_-1)	-0.038 (Δ³y_-1)
4.0	1	18.644	-1.382	0.199	-
4.5	2	17.262	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Gauss Forward Polynomial Formula:

P(x) = y_0 + p \Delta y_0 + \frac{p(p-1)}{2!} \Delta^2 y_{-1} + \frac{(p+1)p(p-1)}{3!} \Delta^3 y_{-1}

Substituting values ( $p = 0.5, y_0 = 20.225, \Delta y_0 = -1.581, \Delta^2 y_{-1} = 0.237, \Delta^3 y_{-1} = -0.038$ ):

P(3.75) = 20.225 + (0.5)(-1.581) + \frac{(0.5)(-0.5)}{2}(0.237) + \frac{(1.5)(0.5)(-0.5)}{6}(-0.038)

P(3.75) = 20.22500 - 0.79050 - 0.02963 + 0.00238

P(3.75) = 19.40725

FINAL RESULT

Interpolated Value at x = 3.75: P(3.75) = 19.4072

\boxed{P(3.75) = 19.4072}

Using Gauss Forward central difference interpolation, the calculated value at $x = 3.75$ is exactly 19.4072!

LABORATORY

Interactive Live Calculator & Visualizer

CENTRAL DIFFERENCE INTERPOLATOR

Gauss Forward Calculation Engine

PreviousNewton Backward Interpolation

Unit 2•8 of 28

NextGauss Backward Interpolation

Point #	X Value	Y Value = f(X)	Action
P1
P2
P3
P4
P5
P6

Point #	X Value	Y Value = f(X)	Action
P1
P2
P3
P4
P5
P6

Gauss Forward Central Difference Method

Step-by-Step Worked Example

Interpolated Value at x = 3.75: P(3.75) = 19.4072

Interactive Live Calculator & Visualizer

Gauss Forward Calculation Engine

NETZ Engine

Gauss Forward Central Difference Method

Step-by-Step Worked Example

Interpolated Value at x = 3.75: P(3.75) = 19.4072

Interactive Live Calculator & Visualizer

Gauss Forward Calculation Engine

Gauss Forward
Central Difference Method

Gauss Forward
Central Difference Method