Download Advanced Mathematics by C. B. Gupta PDF

By C. B. Gupta

ISBN-10: 8122426859

ISBN-13: 9788122426854

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Example text

Then y = f (x) = a0 (x – x1) (x – x2) ..... (x – xn) + a2(x – x0) (x – x1) ..... (x – xn) ..... + an (x – x0) (x – x1) ..... , an are constants. 7), we get y0 = a0(x0 – x1) (x0 – x2) ...... (x0 – xn) ⇒ a0 = y0 ( x 0 − x1 ) ( x 0 − x 2 ) ...... 7), we get y1 = a1 (x1 – x0) (x1 – x2) ...... (x1 – xn) ⇒ a1 = y1 ( x1 − x 0 ) ( x1 − x 2 ) ...... ( x1 − x n ) Proceeding in this way, we get an = yn ( x n − x 0 ) ( x n − x1 ) ...... 7), we get y = f (x) = ( x − x1 ) ( x − x 2 ) ...... ( x − x n ) y ( x 0 − x1 ) ( x 0 − x 2 ) ......

H2 + (x – a) (x – a – h) (x – a – 2h) ∆3 f ( a) + ...... 3 ! 4) + (x – a) (x – a – h) (x – a – 2h) + ..... + (x – a – (n – 1)h) ∆ f ( a) n ! hn ! 4), we get u(u − 1) 2 u(u − 1) (u − 2) ..... (u − ( n − 1)) n f (a + hu) = f (a) + u∆ f (a) + ∆ f (a) + ..... + ∆ f (a). 2! n! 3. , a + nh, respectively where h is the difference of arguments. Let f (x) be a polynomials in x of degree n. So f (x) can be written as f (x) = a0 + a1(x – a – nh) + a2(x – a – nh) (x – a – (n – 1)h) + a3 (x – a – nh) (x – a – (n – 1)h) (x – a – (n – 2)h) + .....

X − x n ) y + ..... ( x1 − x 0 ) ( x1 − x 2 ) ( x1 − x 3 ) ...... ( x1 − x n ) 1 + ( x − x 0 ) ( x − x1 ) ( x − x 2 ) ...... ( x − x n −1 ) y ( x n − x 0 ) ( x n − x1 ) ( x n − x 2 ) ...... ( x n − x n −1 ) n which is the Lagrange’s interpolation formula. 5. STIRLING’S DIFFERENCE FORMULA The mean of Gauss’s forward difference formula and Gauss’s backward difference formula gives Stirling’s difference formula We have Gauss’s forward difference formula is yu = y0 + u ∆y0 + u(u − 1) 2 (u + 1) u (u − 1) 3 ∆ y–1 + ∆ y–1 2!

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