M3 Numerical Methods — Formula Sheet kuch kuch important topics

 

M3 Numerical Methods — Formula Sheet

1. Newton-Raphson Method

$$\boxed{x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}}$$

Yaad rakho:

$$\boxed{\text{New }x=\text{Old }x-\frac{\text{Function}}{\text{Derivative}}}$$

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2. Regula-Falsi Method

Pehle interval $(a,b)$ choose karo such that:

$$\boxed{f(a)f(b)<0}$$

Formula:

$$\boxed{x=\frac{a f(b)-b f(a)}{f(b)-f(a)}}$$

Equivalent form:

$$\boxed{x=a-\frac{f(a)(b-a)}{f(b)-f(a)}}$$

Phir $f(x)$ ka sign check karke $a$ ya $b$ replace karte hain.

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3. Newton Forward Interpolation

Kab? Jab required $x$, table ke beginning ke paas ho.

$$\boxed{ y= y_0 +u\Delta y_0 +\frac{u(u-1)}{2!}\Delta^2y_0 +\frac{u(u-1)(u-2)}{3!}\Delta^3y_0+\cdots }$$

Where:

$$\boxed{u=\frac{x-x_0}{h}}$$

and

$$h=x_1-x_0$$

Pattern:

$$u,\quad u(u-1),\quad u(u-1)(u-2)$$

Matlab Forward = minus pattern.

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4. Newton Backward Interpolation

Kab? Jab required $x$, table ke end ke paas ho.

$$\boxed{ y= y_n +u\nabla y_n +\frac{u(u+1)}{2!}\nabla^2y_n +\frac{u(u+1)(u+2)}{3!}\nabla^3y_n+\cdots }$$

Where:

$$\boxed{u=\frac{x-x_n}{h}}$$

Pattern:

$$u,\quad u(u+1),\quad u(u+1)(u+2)$$

Matlab Backward = plus pattern.

🔥 Forward vs Backward trick

$$\boxed{\text{Forward → beginning → }u-1,u-2}$$ $$\boxed{\text{Backward → end → }u+1,u+2}$$

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5. Trapezoidal Rule

Given:

$$\int_a^b f(x)\,dx$$

Step size:

$$\boxed{h=\frac{b-a}{n}}$$

Formula:

$$\boxed{ I=\frac{h}{2} [ y_0+y_n+ 2(y_1+y_2+\cdots+y_{n-1}) ] }$$

Coefficient pattern

$$\boxed{1,\;2,\;2,\;2,\ldots,\;2,\;1}$$

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6. Simpson's 1/3 Rule

Condition:

$$\boxed{n=\text{even}}$$

Step size:

$$\boxed{h=\frac{b-a}{n}}$$

Formula:

$$\boxed{ I= \frac{h}{3} [ y_0+y_n +4(y_1+y_3+y_5+\cdots) +2(y_2+y_4+y_6+\cdots) ] }$$

⭐ Coefficient pattern

$$\boxed{1,\;4,\;2,\;4,\;2,\;4,\ldots,\;1}$$

Easy trick:

Odd positions → 4
Even positions → 2

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7. Simpson's 3/8 Rule

Condition:

$$\boxed{n=\text{multiple of 3}}$$

Like:

$$n=3,6,9,12,\ldots$$

Step size:

$$\boxed{h=\frac{b-a}{n}}$$

Formula:

$$\boxed{ I= \frac{3h}{8} [ y_0+y_n+ 3(y_1+y_2+y_4+y_5+\cdots) + 2(y_3+y_6+y_9+\cdots) ] }$$

⭐ Coefficient pattern

$$\boxed{1,\;3,\;3,\;2,\;3,\;3,\;2,\ldots,\;1}$$

Easy trick:

Every third interior point → 2
Remaining interior points → 3

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8. Runge-Kutta Fourth Order — First Order

Given differential equation:

$$\boxed{\frac{dy}{dx}=f(x,y)}$$

We calculate four values:

$$\boxed{k_1=hf(x_n,y_n)}$$ $$\boxed{ k_2= hf\left( x_n+\frac h2, y_n+\frac{k_1}{2} \right) }$$ $$\boxed{ k_3= hf\left( x_n+\frac h2, y_n+\frac{k_2}{2} \right) }$$ $$\boxed{ k_4= hf(x_n+h,y_n+k_3) }$$

Finally:

$$\boxed{ y_{n+1} = y_n+ \frac{1}{6} (k_1+2k_2+2k_3+k_4) }$$

🔥 RK4 memory trick

Remember:

$$\boxed{1,\;2,\;2,\;1}$$

because final formula:

$$\frac{k_1+2k_2+2k_3+k_4}{6}$$

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9. Runge-Kutta Fourth Order — Second Order

Suppose:

$$\boxed{\frac{d^2y}{dx^2}=f\left(x,y,\frac{dy}{dx}\right)}$$

Pehle put:

$$\boxed{z=\frac{dy}{dx}}$$

Then second-order equation ko two first-order equations mein convert karo:

$$\boxed{\frac{dy}{dx}=z}$$

and

$$\boxed{\frac{dz}{dx}=f(x,y,z)}$$

Ab RK4 simultaneously dono par lagta hai.

For $y$:

$$k_1=hz_n$$

and for $z$:

$$l_1=hf(x_n,y_n,z_n)$$

Then:

$$k_2=h\left(z_n+\frac{l_1}{2}\right)$$ $$l_2= hf\left( x_n+\frac h2, y_n+\frac{k_1}{2}, z_n+\frac{l_1}{2} \right)$$

Then:

$$k_3=h\left(z_n+\frac{l_2}{2}\right)$$ $$l_3= hf\left( x_n+\frac h2, y_n+\frac{k_2}{2}, z_n+\frac{l_2}{2} \right)$$

Then:

$$k_4=h(z_n+l_3)$$ $$l_4= hf(x_n+h,y_n+k_3,z_n+l_3)$$

Finally:

$$\boxed{ y_{n+1} = y_n+ \frac{k_1+2k_2+2k_3+k_4}{6} }$$

and

$$\boxed{ z_{n+1} = z_n+ \frac{l_1+2l_2+2l_3+l_4}{6} }$$

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🧠 One-Page Memory Table

Method	Formula / Main Trick
Newton-Raphson	$\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$
Regula-Falsi	$\displaystyle x=\frac{af(b)-bf(a)}{f(b)-f(a)}$
Newton Forward	$u=(x-x_0)/h$, minus pattern
Newton Backward	$u=(x-x_n)/h$, plus pattern
Trapezoidal	$\frac h2[1,2,2,\ldots,2,1]$
Simpson 1/3	$\frac h3[1,4,2,4,2,\ldots,1]$, $n$ even
Simpson 3/8	$\frac{3h}{8}[1,3,3,2,3,3,2,\ldots,1]$, $n$ multiple of 3
RK4 First Order	$k_1,k_2,k_3,k_4$, final weights 1,2,2,1
RK4 Second Order	Put $z=y'$, then use $k$'s + $l$'s

⭐ Sabse important formulas

Exam ke liye agar priority banaoge, to especially ye patterns pakke rakho:

$$\boxed{\text{Newton-Raphson: Function/Derivative}}$$ $$\boxed{\text{Forward: }-1,-2,-3}$$ $$\boxed{\text{Backward: }+1,+2,+3}$$ $$\boxed{\text{Trapezoidal: }1,2,2,2,1}$$ $$\boxed{\text{Simpson 1/3: }1,4,2,4,2,1}$$ $$\boxed{\text{Simpson 3/8: }1,3,3,2,3,3,1}$$ $$\boxed{\text{RK4: }1,2,2,1}$$

Next best step is har method ka ek RGPV-level numerical solve karna, kyunki formula dekhna easy hai but exam mein question se $h,n,u,k_1,k_2$ etc. kaise nikalte hain wahi main part hai.