Normalized 1D General Quadrature Scheme Problem Point-Weight Selection Criteria Result should be exact if f(a) is a polynomia f(x)=a+a1+a2x2+…+a1a=p(x) Select a, 's and w,'s such that p()d=∑op( 0 for ANY polynomial upto(and including)tth order With 2n degrees of freedom, I= 2n-1 SMA-HPC(2002 MIT Numerical Quadrature 9
Normalized 1D General Quadrature Scheme Problem Why the Exactness Criteria? Consider the Taylor series for f(ec) af(0) 14f(0) f(x)=f(0)+ 2+∴…+ l!2l2+B+1 R+1 is the remainder R1+1 8+1f() (+1)!ah24 Whee∈[0,2 SMA-HPC(2002 MIT Numerical Quadrature 10
Normalized 1D General Quadrature Scheme Problem Estimating the Error Using the Taylor series results and the exactness criteria 0+1f(a(c) f()dn-t=1:(c)= do (1+1)!0a4+1 Remainder Assuming derivatives of f(e)are bounded on 0, 1 f(a)da-∑f(m 0 =1 (+1)! Convergence is very fast !! SMA-HPC(2002 MIT Numerical Quadrature 11