UNIT 2 • EQUAL INTERVAL INTERPOLATION

Gauss Backward
Central Difference Method

When your target point lies just past or near the center of a table, Gauss Backward Interpolation starts at a central origin point $x_0$ and steps backwards first, offering maximum accuracy for middle-to-lower values!

GAUSS BACKWARD FORMULACENTRAL ANCHOR

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

1. Central Fraction (p)

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

(distance from central origin

x_0

2. Upward Step FirstUses

\Delta y_{-1}

, then

\Delta^2 y_{-1}

, then

\Delta^3 y_{-2}

3. Equal Mesh Step (h)Uniform spacing between table x entries:

h = x_1 - x_0

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the population dataset (in millions):

Year (x)	1931	1941	1951	1961	1971
Population (y)	15	20	27	39	52

Let's estimate the population for the year $x = 1946$ !

STEP 01 • SELECT CENTRAL ORIGIN & CALCULATE p

We pick $x_0 = 1951$ as our central origin because $1946$ lies just before it. With step size $h = 10$ :

p = \frac{x - x_0}{h} = \frac{1946 - 1951}{10} = \frac{-5}{10} = -0.5

STEP 02 • CONSTRUCT CENTRAL DIFFERENCE TABLE

Year (x)	p	y	Δy	Δ²y	Δ³y
1931	-2	15	-	-	-
1941	-1	20	7 (Δy_-1)	2 (Δ²y_-2)	-
1951 (x_0)	0	27 (y_0)	12 (Δy_0)	5 (Δ²y_-1)	3 (Δ³y_-2)
1961	1	39	13	1	-4
1971	2	52	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Gauss Backward Polynomial Formula:

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

Substituting our calculated values ( $p = -0.5, y_0 = 27, \Delta y_{-1} = 7, \Delta^2 y_{-1} = 5, \Delta^3 y_{-2} = 3$ ):

P(1946) = 27 + (-0.5)(7) + \frac{(0.5)(-0.5)}{2}(5) + \frac{(0.5)(-0.5)(-1.5)}{6}(3)

P(1946) = 27.00000 - 3.50000 - 0.62500 + 0.18750

P(1946) = 23.06250 \text{ million}

FINAL RESULT

Interpolated Population (1946): P(1946) = 23.0625 Million

\boxed{P(1946) = 23.0625 \text{ million}}

Using Gauss Backward central difference interpolation, the estimated population in 1946 is 23.0625 million!

LABORATORY

Interactive Live Calculator & Visualizer

CENTRAL DIFFERENCE INTERPOLATOR

Gauss Backward Calculation Engine

Point #	X Value	Y Value = f(X)	Action
P1

PreviousGauss Forward Interpolation

Unit 2•9 of 28

NextLagrange Interpolation

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 2 • EQUAL INTERVAL INTERPOLATION

Gauss Backward
Central Difference Method

GAUSS BACKWARD FORMULACENTRAL ANCHOR

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

1. Central Fraction (p)

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

(distance from central origin

x_0

2. Upward Step FirstUses

\Delta y_{-1}

, then

\Delta^2 y_{-1}

, then

\Delta^3 y_{-2}

3. Equal Mesh Step (h)Uniform spacing between table x entries:

h = x_1 - x_0

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the population dataset (in millions):

Year (x)	1931	1941	1951	1961	1971
Population (y)	15	20	27	39	52

Let's estimate the population for the year $x = 1946$ !

STEP 01 • SELECT CENTRAL ORIGIN & CALCULATE p

We pick $x_0 = 1951$ as our central origin because $1946$ lies just before it. With step size $h = 10$ :

p = \frac{x - x_0}{h} = \frac{1946 - 1951}{10} = \frac{-5}{10} = -0.5

STEP 02 • CONSTRUCT CENTRAL DIFFERENCE TABLE

Year (x)	p	y	Δy	Δ²y	Δ³y
1931	-2	15	-	-	-
1941	-1	20	7 (Δy_-1)	2 (Δ²y_-2)	-
1951 (x_0)	0	27 (y_0)	12 (Δy_0)	5 (Δ²y_-1)	3 (Δ³y_-2)
1961	1	39	13	1	-4
1971	2	52	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Gauss Backward Polynomial Formula:

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

Substituting our calculated values ( $p = -0.5, y_0 = 27, \Delta y_{-1} = 7, \Delta^2 y_{-1} = 5, \Delta^3 y_{-2} = 3$ ):

P(1946) = 27 + (-0.5)(7) + \frac{(0.5)(-0.5)}{2}(5) + \frac{(0.5)(-0.5)(-1.5)}{6}(3)

P(1946) = 27.00000 - 3.50000 - 0.62500 + 0.18750

P(1946) = 23.06250 \text{ million}

FINAL RESULT

Interpolated Population (1946): P(1946) = 23.0625 Million

\boxed{P(1946) = 23.0625 \text{ million}}

Using Gauss Backward central difference interpolation, the estimated population in 1946 is 23.0625 million!

LABORATORY

Interactive Live Calculator & Visualizer

CENTRAL DIFFERENCE INTERPOLATOR

Gauss Backward Calculation Engine

Point #	X Value	Y Value = f(X)	Action
P1

PreviousGauss Forward Interpolation

Unit 2•9 of 28

NextLagrange Interpolation

Gauss Backward Central Difference Method

Step-by-Step Worked Example

Interpolated Population (1946): P(1946) = 23.0625 Million

Interactive Live Calculator & Visualizer

Gauss Backward Calculation Engine

NETZ Engine

Gauss Backward Central Difference Method

Step-by-Step Worked Example

Interpolated Population (1946): P(1946) = 23.0625 Million

Interactive Live Calculator & Visualizer

Gauss Backward Calculation Engine

Gauss Backward
Central Difference Method

Gauss Backward
Central Difference Method