By S. Kusuoka, A. Yamazaki

A lot of financial difficulties can formulated as restricted optimizations and equilibration in their ideas. numerous mathematical theories were offering economists with quintessential machineries for those difficulties bobbing up in financial conception. Conversely, mathematicians were inspired by means of numerous mathematical problems raised through financial theories. The sequence is designed to collect these mathematicians who have been heavily drawn to getting new demanding stimuli from financial theories with these economists who're looking for potent mathematical instruments for his or her researchers.

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A3 n x n ] a33 . : xn = 1 [bn − a n1 x1 − a n 2 x 2 ...... S. 6), we get x1( 2 ) = 1 [b1 − a12 x 2(1) − a13 x3(1) ...... a1n x n(1) ] a11 45 SOLUTION OF LINEAR SIMULTANEOUS EQUATIONS Now put x1( 2 ) x 2(1) x 3(1) ...... S. 6), we get x 2( 2 ) = 1 [b2 − a21 x1( 2 ) − a23 x 3(1) ...... 6), we get x 3( 2 ) = 1 [ b3 − a31 x1( 2 ) − a32 x 2( 2 ) ...... 6), we get 1 [ bn − an1 x1( 2 ) − an 2 x 2( 2 ) ...... , xn are obtained to desired accuracy Gauss-Seidel method is also known as a method of successive displacement.

Given u0 = 580, u1 = 556, u2 = 520, u3 = —, u4 = 384, find u3. Solution. Let the missing term u3 = X 26 ADVANCED MATHEMATICS ∴ The forward difference table is x ux 0 580 1 556 ∆u x ∆2u x ∆ 3u x ∆ 4 ux – 24 – 12 – 36 2 X – 472 520 X – 484 1860 – 4X X – 520 3 1388 – X X 904 – 2X 384 – X 4 384 Here four values of ux are given. Therefore, we can assume ux to be a polynomial of degree 3 in x ∴ ∆ 4 ux = 0 or or 1860 – 4X = 0 X = 465. Aliter: Here four values of ux are given. Therefore, we can assume ux to be a polynomial of degree 3 in x ∴ ∆ 4 ux = 0 or (E – I)4 ux = 0 or (E4 – 4C1 E3 + 4C2E2 – 4C3 E + I) ux = 0 or E4 ux – 4E3 ux + 6E2 ux – 4Eux + ux = 0 or ux + 4h – 4ux + 3h + 6ux + 2h – 4ux + h + ux = 0 Putting x = 0 and h = 1, we get u4 – 4u3 + 6u2 – 6u1 + u0 = 0 or 384 – 4X + 6 × 520 – 6 × 556 + 580 = 0 1860 – 4X = 0 ⇒ X = 465.

3! + (u + 1) u (u − 1) (u − 2) 4 ∆ y–2 + ...... 4! 8) and Gauss’s backward difference formula is yu = y0 + u∆y– 1 + (u + 2) (u + 1) u (u − 1) u (u + 1) 2 (u + 1) u (u − 1) 3 ∆ y–1 + ∆ y–2 + ∆4y–2 + ...... 4! 2! 3! 9) is 3 3 u(u 2 − 1) ( ∆ y− 1 + ∆ y−2 ) u 2 (u 2 − 1) 4 ( ∆y0 + ∆y−1 ) u 2 2 ∆ y + + ∆ y–2 + ..... + –1 3! 2 4! 2 2! This formula is called the Stirling’s difference formula. yu = y0 + u SOLVED EXAMPLES Example 1. Given u0 = 580, u1 = 556, u2 = 520, u3 = —, u4 = 384, find u3. Solution.