(50G) Ver 6 Aitken-Steffensen Delta-2 & Newton's algorithms +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (https://www.hpmuseum.org/forum/forum-10.html) +--- Forum: General Software Library (https://www.hpmuseum.org/forum/forum-13.html) +--- Thread: (50G) Ver 6 Aitken-Steffensen Delta-2 & Newton's algorithms (/thread-24979.html)

(50G) Ver 6 Aitken-Steffensen Delta-2 & Newton's algorithms - Gil - 2026-04-21 17:28 1) Aitken-Steffensen algorithm Suppose that you have to solve  X=f(X)='0.01+0.9*SIN(X)'. Put, in stack level 2, the expression  '0.01+0.9*SIN(X)'  <containing just the variable X>.  Put then, in stack level 1, an approximate value of X, say 0.01. Run Aitken program.  New Aitken-Steffensen code  \<< "Sol Aitken-Steffensen X=f(X) (not f(X)=0)  2 ARG : .f [in Stack 2] .initial guess # Xo for X [in Stack 1] .Xo & X better not complex # \|>  f  is expr: '  X  ' \|>  f  with var  X  (no other var/letter) \|>  f  may be    NOT   differentiable  " DROP DTAG \-> f X   \<< 'X' 'f' RCL = 'f(X)' 'f' RCL = DEFINE X "With Xo" \->TAG 1. 200.     FOR i 'f(X)' \->NUM 'f(f(X))' \->NUM 'f(X)' \->NUM - DUP ABS .000000000001 <       IF       THEN DROP2 X "X" \->TAG 'f(X)' \->NUM X - "f(X)-X" \->TAG i 1. - "Steps 3*" \->TAG UNROT KILL       END INV 'f(X)' \->NUM X - INV - DUP ABS .000000000001 <       IF       THEN DROP2 X "X" \->TAG 'f(X)' \->NUM X - "f(X)-X" \->TAG i 1. - "Steps 3*" \->TAG UNROT KILL       END INV + 'X' STO     NEXT X "last X  NOT   sol " \->TAG 'f(X)' \->NUM X - "f(X)-X" \->TAG   \>> \>> 2) Just, as comparison, Newton's algorithm Put here, in stack level 2, the modified expression  '0.1+0.9*SIN(X) —X'  <that must be equal to zero> and again, in stack level 1, the same approximate value of X, ie 0.01. Run Newton program.  Corrected code for Newton's program \<< "Sol  f(X)=0 (not X=f(X))  2 ARG : .f [in Stack 2] .initial guess #  for X or z [Stack 1]   \|>  f  with var  X  (no other var/letter) \|>  f  like:   . ' X ' or   . \<< \-> X ' X' \>>  or   . \<< \-> X \<< X \>> \>> or   . \<<\>> \|> inside '' or \<<\>> =0 \|>  f  may be    NOT   differentiable  " DROP DTAG DUP2 1. * DUP IM 0. == { RE } IFT DUP TYPE 1. ==   IF   THEN DUP RE   ELSE DUP   END ABS LOG IP 9. - ALOG 0. 0. 999. 0. RCLF .00000000001 \-> f \<-X h \<-n f\180 max \<-s \<-fg \Ge   \<< "With Xo" \->TAG -22. CF -103. SF -3. CF -105. SF 'f' RCL TYPE 9. ==     IF     THEN 'f(X)' 'f' RCL = DEFINE     END     DO 1. '\<-n' STO+ \<-X h + f       IFERR EVAL \->NUM       THEN DROP DTAG "+ h " msg       END \<-X h - f       IFERR EVAL \->NUM       THEN DROP2 DTAG "- h " msg       END - \->NUM h / 2 . / DUP 'f\180' STO 0. ==       IF       THEN \<-n 1. - R\->I \<-X DUP f \->NUM DUP ABS \Ge >         IF         THEN ROT " Stop  at n" \->TAG UNROT SWAP "& no sol; last X"         ELSE ROT "Iterations n" \->TAG UNROT SWAP " Solution X "         END \->TAG SWAP "f(X)" \->TAG \<-fg STOF DUP ABS \Ge >         IF         THEN 0. R\->I "as  f ' (X) " \->TAG " f ' (X) = 0 : Impossible to continue Newton's alg" DOERR         ELSE KILL         END       END '\<-X-f(\<-X)/f\180'       IFERR EVAL \->NUM       THEN 3. DROPN DTAG "" msg       END DUP \<-X - ABS SWAP '\<-X' STO \Ge < \<-n max == OR f\180 0. == OR     UNTIL     END \<-n R\->I "Iterations" \<-n max == "=max" "" IFTE + \->TAG \<-X \<-n max ==     IF     THEN ROT DTAG UNROT "No sol, last X"     ELSE "Solution X"     END \->TAG DUP f \->NUM "f(X)" \->TAG \<-fg STOF   \>> \>> 3) When dealing with converging series above all alternate series, Aitk2 simple program — that uses Aitken Delta-2 — might help find, with three given partial sums S[n], S[n+1] and S[n+2], an approximate value of the final result (S[infinity]).  Example ln(2) = sum from n=1 to n=infinity of (-1)^(n+1)/n. Let's calculate the partial sum S1, S2 and S3: 1 for S1,  .5 for S2,  .833333333333 for S3,  and put the above found values respectively in stack level 3, 2 and 1. Run Aitk2, to find 0.7 (vs 0.693147, the real value), a much better value than the initial given partial sums S1, S2 and S3.  If we want a still much better value than 0.7, we could calculate for instance S100, S101 and S102: .688172179304 for S100,  .698073169403 for S101,  .688269247834 for S102,  and put again those values respectively in stack level 3, 2 and 1. Run Aitk2, to find now 0.693147.  Note that, to get the same value  (0.693147) with partial sums, we should go up to S[1000 000], this example showing the interest of that very simple method.  Code Aitk2  \<< "3 Arg/partial sums: S[n] S[n+1] S[i+2] of conv. series works better in oscillat.series " DROP \-> S0 S1 S2   \<< '(S0*S2-S1^2)/(S0-2*S1+S2)' \->NUM "A[n]" \->TAG   \>> \>> Idea of the program  when reading the  reference: https://fr.wikipedia.org/wiki/Delta-2#/languages given by Albert Chan in one of his last posts. Thanks to Thomas Klemm  for having spotted an import error in Newton's code.