AMME2960 Assignment 1.docx - AMME2960 Assignment 1 Part 1 Q1(i Given that f xxx x=x f i2 B f i1 C f i f i 1 i x3(2 x(2 x)2(2 x)3 f i 2 =f i f x x=x f xx

AMME2960 Assignment 1.docx - AMME2960 Assignment 1 Part 1...

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AMME2960 Assignment 1Part 1Q1(i):Given that:fxxxx=xifi2+Bfi1+C fi+fi+1(∆ x3)+εfi2=fi+(2∆ x)1!fx¿x=xi+(2∆x)22!fxx¿x=xi+(2∆ x)33!fxxx¿x=xi+fi1=fi+(1∆ x)1!fx¿x=xi+(1∆ x)22!fxx¿x=xi+(1∆x)33!fxxx¿x=xi+fi=fifi+1=fi+(∆x)1!fx¿x=xi+(∆ x)22!fxx¿x=xi+(∆ x)33!fxxx¿x=xi+Taylor series:f¿i+1(1)f¿i+D1¿f¿i1+C1¿f¿i2+B1¿fxxx=A1¿therefore, combining the above and the four equationsA1fi2=A1fi+A1(2∆ x)1!fx¿x=xi+A1(2∆ x)22!fxx¿x=xi+A1(2∆ x)33!fxxx¿x=xi+(2)B1fi1=B1fi+B1(∆x)1!fx¿x=xi+B1(∆ x)22!fxx¿x=xi+B1(2∆ x)33!fxxx¿x=xi+(3)C1fi=C1fi(4)D1fi1=D1fi+D1(∆x)1!fx¿x=xi+D1(∆ x)22!fxx¿x=xi+D1(∆ x)33!fxxx¿x=xi+(5)(2) + (3) + (4) + (5) equate to (1) coefficient of coefficient of fA1+B1+C1+D1=0(6)(2) + (3) + (4) + (5) equate to (1) coefficient of coefficient of fx
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2A1B1+D1=0(7)(2) + (3) + (4) + (5) equate to (1) coefficient of coefficient of fxx4A1+B1+D1=0(8)(6) + (7)A1+C1+D1=0(9)(8) – (7)6A1+2B1=03A1+B1=03A1=B1(10)(2) + (3) + (4) + (5) equate to (1) coefficient of coefficient of fxxxA1(2∆ x)33!+B1(1∆ x)33!+D1(∆ x)33!=1(11)8A1B1+D1=3!(∆ x)3=6(∆ x)3(12)From (8) 4A1B1=D18A1B1B14A1=6(∆ x)312A12B1=6(∆ x)36A1B1=3(∆ x)3(13)(10) (13)6A13A1=3(∆ x)3A1=1(∆ x)3(14)(14) (10) B1=3(∆ x)3(15)(14) & (15) (7)D1=2A1+B1=3(∆ x)32(∆ x)3=1(∆ x)3(16)(14), (15), (16) (6)C1=−A1B1D1=−(1(∆ x)3+1(∆ x)3+3(∆ x)3)=3(∆ x)3(17)Therefore, utilising equations (15) & (17) and comparing equation (1) and given equation
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∆ x¿3B¿∆ x¿3×3(∆x)3=3B=¿¿∆ x¿3C¿∆ x¿3×3(∆x)3=−3C=¿¿B = 3 & C = -3Q1(ii):fxxxx=xifi2+3fi13fi+fi+1(∆ x3)+ε(2) + (3) + (4) + (5)3¿2+11¿2+0(¿)fxx¿x=xi¿3¿3+11¿3+0(¿)fxxx¿x=xi¿3¿4+11¿4+0(¿)fxxxx¿x=xi¿2¿4+3¿¿∆ x4¿2¿3+3¿¿∆ x3¿2¿2+3¿¿∆ x2¿¿(1+33+1)fi(∆x)3+∆ x(2+3(1)+0(3)+1)fx¿x=xi1!×∆ x3+¿
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3¿4+11¿4+0(¿)fxxxx¿x=xi¿2¿4+3¿¿∆x1¿¿0fi+0fx¿x=xi+0fxx¿x=xi+fxxx¿x=xi+¿¿fxxx¿x=xi∆ xfxxxx¿x=xi2+leading order term: ¿∆ x fxxxx¿x=xi2Q2Q3
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From the graph it can be demonstrated that as the ∆ xincreases, the error norm also increases.
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