UNIT 2 • EQUAL INTERVAL INTERPOLATION

Newton Forward
Interpolation Method

When you need to estimate a value near the beginning of an equally spaced table, Newton Forward Interpolation anchors at the top row and steps forward into future values!

FORWARD DIFFERENCE FORMULASTART-POINT ANCHOR

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

1. Distance Fraction (v)

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

(distance forward from first entry

x_0

2. Equal Step Size (h)Uniform spacing between table x entries:

h = x_1 - x_0

3. Top-Row Operators (Δ)Uses

\Delta y_0, \Delta^2 y_0, \dots

from the top diagonal.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the quadratic dataset generated by $y = x^2 + 1$ :

x	1	2	3	4
y	2	5	10	17

Let's estimate the value at $x = 2.5$ !

STEP 01 • CALCULATE FORWARD FRACTION (v)

Using initial point $x_0 = 1$ and step size $h = 1$ :

v = \frac{x - x_0}{h} = \frac{2.5 - 1}{1} = 1.5

STEP 02 • CONSTRUCT FORWARD DIFFERENCE TABLE

x	y	Δy	Δ²y	Δ³y
1 (x_0)	2 (y_0)	3 (Δy_0)	2 (Δ²y_0)	0 (Δ³y_0)
2	5	5	2	-
3	10	7	-	-
4	17	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Newton Forward Polynomial Formula:

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

Substituting values ( $v = 1.5, y_0 = 2, \Delta y_0 = 3, \Delta^2 y_0 = 2, \Delta^3 y_0 = 0$ ):

P(2.5) = 2 + (1.5)(3) + \frac{1.5(1.5-1)}{2}(2) + 0

P(2.5) = 2.00000 + 4.50000 + 0.75000 + 0.00000

P(2.5) = 7.25000

FINAL RESULT

Interpolated Value at x = 2.5: P(2.5) = 7.25

\boxed{P(2.5) = 7.25} \quad (\text{Exact: } 2.5^2 + 1 = 7.25)

Our Newton Forward polynomial estimate of 7.25 matches the exact quadratic value $2.5^2 + 1 = 7.25$ 100%!

LABORATORY

Interactive Live Calculator & Visualizer

Newton Forward Interpolation Calculator

Plot:

PreviousGauss-Seidel Method

Unit 2•6 of 28

NextNewton Backward Interpolation

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 2 • EQUAL INTERVAL INTERPOLATION

Newton Forward
Interpolation Method

When you need to estimate a value near the beginning of an equally spaced table, Newton Forward Interpolation anchors at the top row and steps forward into future values!

FORWARD DIFFERENCE FORMULASTART-POINT ANCHOR

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

1. Distance Fraction (v)

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

(distance forward from first entry

x_0

2. Equal Step Size (h)Uniform spacing between table x entries:

h = x_1 - x_0

3. Top-Row Operators (Δ)Uses

\Delta y_0, \Delta^2 y_0, \dots

from the top diagonal.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the quadratic dataset generated by $y = x^2 + 1$ :

x	1	2	3	4
y	2	5	10	17

Let's estimate the value at $x = 2.5$ !

STEP 01 • CALCULATE FORWARD FRACTION (v)

Using initial point $x_0 = 1$ and step size $h = 1$ :

v = \frac{x - x_0}{h} = \frac{2.5 - 1}{1} = 1.5

STEP 02 • CONSTRUCT FORWARD DIFFERENCE TABLE

x	y	Δy	Δ²y	Δ³y
1 (x_0)	2 (y_0)	3 (Δy_0)	2 (Δ²y_0)	0 (Δ³y_0)
2	5	5	2	-
3	10	7	-	-
4	17	-	-	-

STEP 03 • FORMULA & NUMERICAL SUBSTITUTION

Newton Forward Polynomial Formula:

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

Substituting values ( $v = 1.5, y_0 = 2, \Delta y_0 = 3, \Delta^2 y_0 = 2, \Delta^3 y_0 = 0$ ):

P(2.5) = 2 + (1.5)(3) + \frac{1.5(1.5-1)}{2}(2) + 0

P(2.5) = 2.00000 + 4.50000 + 0.75000 + 0.00000

P(2.5) = 7.25000

FINAL RESULT

Interpolated Value at x = 2.5: P(2.5) = 7.25

\boxed{P(2.5) = 7.25} \quad (\text{Exact: } 2.5^2 + 1 = 7.25)

Our Newton Forward polynomial estimate of 7.25 matches the exact quadratic value $2.5^2 + 1 = 7.25$ 100%!

LABORATORY

Interactive Live Calculator & Visualizer

Newton Forward Interpolation Calculator

Plot:

PreviousGauss-Seidel Method

Unit 2•6 of 28

NextNewton Backward Interpolation

Newton Forward Interpolation Method

Step-by-Step Worked Example

Interpolated Value at x = 2.5: P(2.5) = 7.25

Interactive Live Calculator & Visualizer

Newton Forward Interpolation Calculator

Plot:

NETZ Engine

Newton Forward Interpolation Method

Step-by-Step Worked Example

Interpolated Value at x = 2.5: P(2.5) = 7.25

Interactive Live Calculator & Visualizer

Newton Forward Interpolation Calculator

Plot:

Newton Forward
Interpolation Method

Newton Forward
Interpolation Method