DO starting points kyun? Kyunki ek line define karne ke liye do points chahiye. Newton ko ek point aur ek slope chahiya tha; hum slope ki jagah ek doosra point le lete hain.
Hum formula seedha nahi thopenge. Chalte hain build karte hain.
Step 1 — Secant line likho. Curve pe do points hain:
Pn−1=(xn−1,f(xn−1)),Pn=(xn,f(xn)).Yeh step kyun? Yeh do sabse recent guesses hain — hum latest information use karte hain.
Step 2 — Chord ka slope. Unse guzarne wali line ka slope m hai:
m=xn−xn−1f(xn)−f(xn−1).Yeh step kyun? Yeh secant ka rise-over-run hai — yeh f′(xn) ko approximate karta hai. (Newton se compare karo, jo exact f′(xn) use karta hai.)
Step 3 — Line ki equationPn se guzarti hui slope m ke saath:
y−f(xn)=m(x−xn).
Step 4 — Yeh x-axis ko kahan cross karti hai?y=0 set karo aur use x=xn+1 bolte hain:
0−f(xn)=m(xn+1−xn).Yeh step kyun? Root wahan hota hai jahan f=0 ho; line ka x-intercept us root ke liye hamara best linear guess hai.
Step 5 — xn+1 ke liye solve karo:xn+1−xn=−mf(xn)=−f(xn)−f(xn−1)f(xn)(xn−xn−1).xn+1=xn−f(xn)f(xn)−f(xn−1)xn−xn−1
Koi acha sa derivative nahi chahiye. x0=0,x1=1 lo.
f(0)=1,f(1)=cos1−1=−0.4597.x2=1−(−0.4597)⋅−0.4597−11−0=1−(−0.4597)(−0.6850)=0.6851.Yeh step kyun? Same recipe; Secant ko koi farak nahi padta ki cosx−x ka koi simple closed-form root nahi hai.
Aage chalte hain toh x3≈0.7363, x4≈0.7391 milta hai — 0.73909 pe converge ho raha hai.
Socho ek pahaad hai jahan height exactly zero hai uss jagah jo tumhe dhundhni hai. Tum do jagah khade ho aur donou jagah apni height naapte ho. Ek seedha ruler rakho jo un donou heights ko jode. Jahan ruler zameen (zero height) ko chhoye woh tumhara agla guess hai — wahaan jao, apne do spots mein se purana wala chhod do, phir napo, aur naya ruler rakho. Har ruler seedha aur magic zero-spot ke paas hota hai, toh thodi koshishon mein tum basically wahaan pahunch jaate ho. Clever part yeh hai: tumhe kabhi nahi jaanna ki pahaad kitna steep hai — bas do heights kaafi hain.
Nahi — yeh use finite difference (chord slope) se approximate karta hai.
Secant method ki order of convergence kya hai?
Golden ratio, p=21+5≈1.618 (superlinear).
Secant vs Newton: kya replace hota hai?
f′(xn) ki jagah xn−xn−1f(xn)−f(xn−1) aata hai.
Kya Secant method ek bracketing method hai?
Nahi, yeh ek open method hai — koi guaranteed convergence nahi aur iterate interval se bahar ja sakta hai.
Secant aur Regula Falsi mein kya farak hai?
Secant hamesha do latest points use karta hai (order 1.618, no bracketing); Regula Falsi ek aisa point rakhta hai jo root bracket karta hai (bounded lekin sirf linear).
Secant, Newton se per evaluation zyada efficient kyun ho sakta hai?
Ise har step mein sirf ek naya f-evaluation chahiye (koi f′ nahi); do Secant steps (1.6182≈2.6) ek Newton step (2) ko beat kar sakte hain.
Secant formula mein numerical blow-up kya cause karta hai?
Denominator f(xn)−f(xn−1)→0 jab consecutive function values almost equal hoon.
