Jeremy Knight Math 639 3/29/11 Algorithm 10.3
<Text-field style="Heading 1" layout="Heading 1">SteepestDescent(n,x0,f,TOL,N)</Text-field> To approximate a solution p to the minimization problem 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 given an initial approximation x. SteepestDescent:=proc(n,x0,f,TOL,N) #n=number of variables, x0 = 1xn array of initial approximation #F:=Array of Functions #TOL=tolerance, N = maximim iterations local a,h,i,j,k,g,g0,G,alpha,alpha0,x,z,z0,L2Norm,Jacobian,gradientG,delG: option trace: ##################### Subroutine Section ##################### g:=proc(x::Array) local i,temp: temp:=0: for i from 1 to n do temp:=temp+(f[i](x))^2: end do: return(temp): end proc: Jacobian:=proc(x::Array) #Approximates the value of the Jacobian at x #F is an array of functions, x is an array of values local i,j,e,h,J: J:=Array(1..n,1..n): e:=Array(1..n): h:=.0001: for i from 1 to n do for j from 1 to n do e[j]:=h: J[i,j]:=(f[i](x+e)-f[i](x-e))/(2*h): e[j]:=0: end do: end do: return(J): end proc: gradientG:=proc(x::Array) local i,j,F,J,temp: #del{g(x)}=2*J(x)^t*F(x); #First calculate the Jacobian J(x) J:=Jacobian(x): delG:=Array(1..n): for i from 1 to n do F[i]:=f[i](x): end do: for i from 1 to n do for j from 1 to n do delG[i]:=delG[i]+2*F[j]*J[j,i]: end do: end do: return(delG): end proc: L2Norm:=proc(x::Array) #calculates the l2 norm of vector x local i,temp: temp:=0: for i from 1 to n do temp:=temp+x[i]^2: end do: temp:=sqrt(temp): return(temp): end proc: ##################### End Subroutine Section ##################### G:=Array(1..3): h:=Array(1..3): alpha:=Array(1..3): k:=1: x:=Array(1..n): for i from 1 to n do x[i]:=x0[i]: end do: while k<=N do G[1]:=g(x): z:=gradientG(x): z0:=L2Norm(z): if z0=0 then print("Zero Gradient"): print(x,G[1]): print("Might have a minimum"): stop: end if: z:=z/z0: alpha[1]:=0: alpha[3]:=1: G[3]:=g(x-alpha[3]*z): while G[3]>=G[1] do alpha[3]:=alpha[3]/2: G[3]:=g(x-alpha[3]*z): if alpha[3]<TOL/2 then print("No likely improvement"): print(x, G[1]): print("Might have a minimum"): stop: end if: end do: alpha[2]:=alpha[3]/2: G[2]:=g(x-alpha[2]*z): #Newton's forward divided-difference formula is used to find #P(alpha)=G[1]+h[1]*alpha+h[3]alpha(alpha-alpha[2]) #interpolates h[alpha] at alpha=0,alpha=alpha[2], alpha= h[1]:=(G[2]-G[1])/alpha[2]: h[2]:=(G[3]-G[2])/(alpha[3]-alpha[2]): h[3]:=(h[3]-h[1])/alpha[3]: #the critical point of P occurs at alpha0: alpha0:=0.5*(alpha[2]-h[1]/h[3]): g0:=g(x-alpha0*z): if g0<G[3] then a:=alpha0: else a:=alpha[3]: end if: x:=x-alpha*z: print("***",k, x , g(x)): if abs(g(x)-G[1])<TOL then print("The Procedure was successful"): print(x, g(x)): stop: end if: k:=k+1: end do: if k>N then print("Maximum iterations exceeded"): end if: end proc:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= f1:=(x::Array) -> 3.0*x[1]-cos(x[2]*x[3])-(1/2); f2:=(x::Array)->x[1]^2-81.0*(x[2]+0.1)^2+sin(x[3])+1.06; f3:=(x::Array)->evalf(exp(-1.0*x[1]*x[2])+20.0*x[3]+(10.0*Pi-3.0)/3.0); f:=Array([f1,f2,f3]); 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjZjNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYjNiUtRiM2JS1GLDYlUSJ4RidGL0YyLUY2Ni1RIzo6RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlotRiw2JVEmQXJyYXlGJ0YvRjItRjY2LVEnJnJhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRllGZW4tRiM2JS1GLDYlUSZldmFsZkYnRi9GMi1GNjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y5RjtGPkZARkJGREZGRkhGWUZlbi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYjNictSSVtc3VwR0YkNiUtRjY2LVEvJkV4cG9uZW50aWFsRTtGJ0Y5RjtGPkZARkJGREZGRkhGWS9GTlEsMC4xMTExMTExZW1GJy1GIzYkLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpcC1GIzYnLUkjbW5HRiQ2JFEkMS4wRidGOS1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIRllGZW4tSSVtc3ViR0YkNiVGUy1GIzYjLUZecTYkUSIxRidGOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRmFxLUZlcTYlRlMtRiM2Iy1GXnE2JFEiMkYnRjlGXHIvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zeci1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRmhwRmpwLUYjNiUtRl5xNiRRJTIwLjBGJ0Y5RmFxLUZlcTYlRlMtRiM2Iy1GXnE2JFEiM0YnRjlGXHJGaHItSSZtZnJhY0dGJDYoLUYjNiMtRiM2JS1GIzYlLUZecTYkUSUxMC4wRidGOUZhcS1GLDYlUSUmcGk7RicvRjBGPUY5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGaHBGanAtRl5xNiRRJDMuMEYnRjktRiM2I0ZqdC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZHUvJSliZXZlbGxlZEdGPUY5 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 x0:=Array([0.0,0.0,0.0]); SteepestDescent(3,x0,f,.0001,10); 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 {--> enter SteepestDescent, args = 3, Array(1..3, {(1) = 0., (2) = 0., (3) = 0.}), Array(1..3, {(1) = f1, (2) = f2, (3) = f3}), 0.1e-3, 10 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