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EQsheet - Romberg R(n,0)= trap rule with 2^n subintervals

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Romberg R(n,0)= trap rule with 2^n subintervals R(n,0)=1/2*R(n-1,0)+h*SUM(k=1,2^(n-1))of f[a+(2k-1)*h] h= (b-a)/2^n R(n,m) = R(n,m-1) +1/(4^m-1)*[R(n,m-1)-R(n-1,m-1)] Original L=L1 Make ur n=n+1 so h becomes h/2 to get L2 Multiply by constant that cancels ur (/2)^a on the first error term > > (2^a)L2 Subtract ur original to cancel out the first error term (2^a-1)L Solve for L by dividing by (2^a-1) Re work eq to get new formula with smaller error term Simpson’s Scheme 1) Integral of f from a to b = approx S(a,b) S(a,b)= ((b-a)/6)[f(a)+4f((a+b)/2)+f(b)] E= (-1/90)[.5(b-a)]^5*f^(4)(e) 2) for h=(b-a)/2 doing the integral from a to a+2h = h/3[f(a)+4(a+h)+f(a+2h)] E=(-1/90)h^5*f^(4)(e) 3) Composite 1/3 over n even subintervals I(a,b)= h/3[f(a)+f(b)] +4h/3S(i=1,n/2)f[a+(2i-1)h] +2h/3S(i=1,(n-2)/2) f(a+2ih) h=(b-a)/n E= -1/180(b-a)h^4f^(4)(e) 4) error test 1/15abs[S(a,c)+S(c,b)-S(a,c)]<e Gaussian Quadrature Formulas I(a,b)=approx S(i=0,n)Aif(xi) Ai=I(a,b)li(x)
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This test prep was uploaded on 04/19/2008 for the course MATH 458 taught by Professor Tuffaha during the Fall '07 term at USC.

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EQsheet - Romberg R(n,0)= trap rule with 2^n subintervals

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