Jeremy Knight Math 639 2/17/11 Algorithm 9.1: Power Method
<Text-field style="Heading 1" layout="Heading 1">PowerMethod(n,A,x,TOL,N,display)</Text-field> PowerMethod:=proc(n,A,x0,TOL,N,display) #n=dimension, A=Array, x=vector, TOL=tolerance, N=max iteraions #This version has been modified to use Aitkens delta^2 procedure local i,j,k,p,x,xp,y,mu,yp,EuclidNorm,ERR,temp,mu0,mu1,muhat: #option trace; k:=1: mu0:=0: mu1:=0: x:=x0: if display=1 then print("iterations, eigenvalue, eigenvector, muhat"); end if; #subroutine for finding the Euclidean Norm and returning the norm and the index of the norm. EuclidNorm:=proc(n,x) #n=vector size, x= 1 by n array local euclid_norm,j,p: euclid_norm:=abs(x[1]): p:=1; for j from 2 to n do if abs(x[j])>euclid_norm then euclid_norm:=abs(x[j]): p:=j: end if: end do: return(euclid_norm,p); end proc: #find the smallest integer p with 1<=p<=n and abs(x[p])=norm(X) xp,p:=EuclidNorm(n,x); #normalize x x:=x/x[p]: while k<=N do #Calculate next eigenvector approximate y:=Array(1..n): for i from 1 to n do for j from 1 to n do y[i]:=y[i]+A[i,j]*x[j]: end do: end do: mu:=y[p]: #use Aitken's delta^2 procedure to speed convergence muhat:=mu0-((mu1-mu0)^2/(mu-2*mu1+mu0)): #find the smallest integer p with 1<=p<=n and abs(y[p])=norm(y) yp,p:=EuclidNorm(n,y): if y[p]=0 then print("Eigenvector",evalf(x)); print("A has the eigenvalue 0, select new vector x and restart"); break; end if: #Modify with abs(x)-abs(y/y[p] to find an accurate error measurement when signs alternate in eigenvector. ERR,temp:=EuclidNorm(n,abs(x)-abs(y/y[p])): x:=y/y[p]; #check for tolerance reached if ERR<TOL and k>=4 then print("The procedure was successful"); print("eigenvalue approximate (muhat)=",evalf(muhat)); print("eigenvector approximate=",evalf(x)); break; end if; if display=1 then print(k, evalf(mu), evalf(x), evalf(muhat)); end if; k:=k+1: mu0:=mu1: mu1:=mu: end do; if k=N then print("The maximum number of iterations exceeded"); end if: if ERR>TOL then print("tolerance not reached"); print("eigenvalue approximate=",evalf(muhat)); print("eigenvector approximate=",evalf(x)); end if; end proc:
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B:=Array([[0.5,1.5],[0,-1]]); x0:=Array([1,1]); 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PowerMethod(2,B,x0,0.00001,20,1); LUkjbXNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2I1FEaXRlcmF0aW9ucyx+ZWlnZW52YWx1ZSx+ZWlnZW52ZWN0b3JGJw== 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 UTZ0b2xlcmFuY2V+bm90fnJlYWNoZWQ2Ig== with(LinearAlgebra): C:=convert(B,Matrix); Eigenvectors(C); 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 solve((0.5-lambda)*(-1-lambda)=0,lambda); NiQkIisrKysrXSEjNSQhIiIiIiE= solve(z+2=9,z); IiIo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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