Introduction

जब data unequally spaced होता है, तब finite difference method काम नहीं करता।

ऐसी स्थिति में हम Divided Differences का उपयोग करते हैं, जो interpolation का एक बहुत important concept है और Newton’s General Interpolation Formula का base भी यही है।

Basic Idea

मान लें data points दिए गए हैं:$$(x_0, y_0), (x_1, y_1), (x_2, y_2), \dots$$

जहाँ:$$y_i = f(x_i)$$

अब हम successive differences निकालते हैं, लेकिन यहाँ difference को $x$ के difference से divide भी किया जाता है

👉 इसी कारण इसे Divided Difference कहा जाता है

First Divided Difference

$$f[x_0, x_1] = \frac{f(x_1) – f(x_0)}{x_1 – x_0}$$

Second Divided Difference

$$f[x_0, x_1, x_2] = \frac{f[x_1, x_2] – f[x_0, x_1]}{x_2 – x_0}$$

Third Divided Difference

$$f[x_0, x_1, x_2, x_3] = \frac{f[x_1, x_2, x_3] – f[x_0, x_1, x_2]}{x_3 – x_0}$$

General Formula

$$f[x_0, x_1, \dots, x_n] = \frac{f[x_1, x_2, \dots, x_n] – f[x_0, x_1, \dots, x_{n-1}]}{x_n – x_0}$$

Divided Difference Table

Example data:

x	y
1	1
2	4
4	16

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Step 1: First Divided Difference

$$f[1,2] = \frac{4-1}{2-1} = 3$$

$$f[2,4] = \frac{16-4}{4-2} = 6$$

Step 2: Second Divided Difference

$$f[1,2,4] = \frac{6-3}{4-1} = 1$$

Table Form

x	y	First Diff	Second Diff
1	1	3	1
2	4	6
4	16

Properties of Divided Differences

1. Symmetry Property

Divided difference symmetric होता है:$$f[x_0, x_1] = f[x_1, x_0]$$

👉 order change करने पर value नहीं बदलती

2. Polynomial Property

यदि $f(x)$ degree $n$ का polynomial है:$$f[x_0, x_1, \dots, x_n] = \text{constant}$$

और$$f[x_0, x_1, \dots, x_{n+1}] = 0$$

3. Relation with Finite Difference

यदि data equally spaced हो, तो:$$f[x_0, x_1] = \frac{\Delta y_0}{h}$$

👉 यानी divided difference finite difference का generalized form है

4. Linearity Property

यदि:$$f(x) = a g(x) + b h(x)$$

तो:$$f[x_0, x_1] = a g[x_0, x_1] + b h[x_0, x_1]$$

5. Zero Property

यदि सभी $y$ values same हैं, तो:$$f[x_0, x_1, \dots] = 0$$

Graphical Understanding

Introduction

जब data unequally spaced होता है, तब finite difference method काम नहीं करता।

ऐसी स्थिति में हम Divided Differences का उपयोग करते हैं, जो interpolation का एक बहुत important concept है और Newton’s General Interpolation Formula का base भी यही है।

Basic Idea

मान लें data points दिए गए हैं:$$(x_0, y_0), (x_1, y_1), (x_2, y_2), \dots$$

जहाँ:$$y_i = f(x_i)$$

अब हम successive differences निकालते हैं, लेकिन यहाँ difference को $x$x के difference से divide भी किया जाता है

👉 इसी कारण इसे Divided Difference कहा जाता है

First Divided Difference

$$f[x_0, x_1] = \frac{f(x_1) – f(x_0)}{x_1 – x_0}$$

Second Divided Difference

$$f[x_0, x_1, x_2] = \frac{f[x_1, x_2] – f[x_0, x_1]}{x_2 – x_0}$$

Third Divided Difference

$$f[x_0, x_1, x_2, x_3] = \frac{f[x_1, x_2, x_3] – f[x_0, x_1, x_2]}{x_3 – x_0}$$

General Formula

$$f[x_0, x_1, \dots, x_n] = \frac{f[x_1, x_2, \dots, x_n] – f[x_0, x_1, \dots, x_{n-1}]}{x_n – x_0}$$

Divided Difference Table

Example data:

x	y
1	1
2	4
4	16

Step 1: First Divided Difference

$$f[1,2] = \frac{4-1}{2-1} = 3$$

$$f[2,4] = \frac{16-4}{4-2} = 6$$

Step 2: Second Divided Difference

$$f[1,2,4] = \frac{6-3}{4-1} = 1$$

Table Form

x	y	First Diff	Second Diff
1	1	3	1
2	4	6
4	16

Properties of Divided Differences

1. Symmetry Property

Divided difference symmetric होता है:$$f[x_0, x_1] = f[x_1, x_0]$$

👉 order change करने पर value नहीं बदलती

2. Polynomial Property

यदि $f(x)$ degree $n$ का polynomial है:$$f[x_0, x_1, \dots, x_n] = \text{constant}$$

और$$f[x_0, x_1, \dots, x_{n+1}] = 0$$

3. Relation with Finite Difference

यदि data equally spaced हो, तो:$$f[x_0, x_1] = \frac{\Delta y_0}{h}$$

👉 यानी divided difference finite difference का generalized form है

4. Linearity Property

यदि:$$f(x) = a g(x) + b h(x)$$

तो:$$f[x_0, x_1] = a g[x_0, x_1] + b h[x_0, x_1]$$

5. Zero Property

यदि सभी $y$ values same हैं, तो:$$f[x_0, x_1, \dots] = 0$$

Graphical Understanding

Graph में:

• points uneven spacing में होते हैं
• दो points के बीच slope निकाला जाता है
• यही divided difference होता है

Importance

• Unequal spacing के लिए essential
• Newton’s General Formula का base
• Lagrange method से related
• Numerical methods में बहुत important

Final Understanding

Divided difference unequal spacing data के लिए difference का generalized form है
यह slope-based concept है
यह Newton interpolation का foundation है
यह exam में long numerical और theory दोनों में पूछा जाता है