Page 187 - Compact Numerical Methods For Computers

176 Compact numerical methods for computers
Algorithm 19. A Nelder-Mead minimisation procedure (cont.)
H := j; VH := f; {save new ‘high’}
end;
end; {if j<>L}
end; {search for highest and lowest}
{STEP 12: test and display current polytope information}
if VH>VL+convtol then
begin {major cycle of the method}
str(funcount:5,tstr); {this is purely for orderly display of results
in aligned columns}
writeln(action,tstr,’‘,VH,’ ‘,VL);
VN := beta*VL+(1.0-beta)*VH;
{interpolate to get “next to highest” function value -- there are
many options here, we have chosen a fairly conservative one.}
for i := 1 to n do {compute centroid of all but point H -- STEP 13}
begin
temp := -P[i,H]; {leave out point H by subtraction}
for j := 1 to nl do temp := temp+P[i,j];
P[i,C] := temp/n; {centroid parameter i}
end, {loop on i for centroid}
for i := 1 to n do {compute reflection in Bvec[] -- STEP 14}
Bvec[i] := (1.0+alpha)*P[i,C]-alpha*P[i,H];
f := fminfn(n,Bvec,Workdata,notcomp); {function value at refln point}
if notcomp then f := big; {When function is not computable, a very
large value is assigned.}
funcount := funcount+l; {increment function count}
action := ‘REFLECTION ’; {label the action taken}
VR := f; {STEP 15: test if extension should be tried}
if VR<VL then
begin {STEP 16: try extension}
P[nl,C] := f; {save the function value at reflection point}
for i := 1 to n do
begin
f := gamma*Bvec[i]+(1-gamma)*P[i,C];
P[i,C] := Bvec[i]; {save the reflection point in case we need it}
Bvec[i] := f;
end; {loop on i for extension point}
f := fminfn(n,Bvec,Workdata,notcomp); {function calculation}
if notcomp then f := big; funcount := funcount+1;
if f<VR then {STEP 17: test extension}
begin {STEP 18: save extension point, replacing H}
for i := 1 to n do P[i,H] := Bvec[i];
P[n1,H] := f; {and its function value}
action := ‘EXTENSION’; {change the action label}
end {replace H}
else
begin {STEP 19: save reflection point}
for i := 1 to n do P[i,H] := P[i,C];
P[n1,H] := VR; {save reflection function value}
end {save reflection point in H; note action is still reflection}
end {try extension}
else {reflection point not lower than current lowest point}
begin {reduction and shrink -- STEP 20}