Creating an equation library (Updated 03-FEB-2017)
01-09-2015, 09:51 PM (This post was last modified: 03-09-2017 04:25 PM by Han.)
Post: #1
 Han Senior Member Posts: 1,883 Joined: Dec 2013
Creating an equation library (Updated 03-FEB-2017)
(This post has been edited; the original content of the first post is at the bottom of this post)

MARCH 2017: This project now exists as an actual app. Please see http://hpmuseum.org/forum/thread-7725.html

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Version 0.063 now makes use of the SVD package I wrote and posted here: http://www.hpmuseum.org/forum/thread-497...67743.html

This version falls back on the SVD package I wrote for certain classes of matrices -- mostly those with singular values of 0. The Prime's implementation computes the left orthonormal vectors based on the reciprocal of the singular values. So when a singular value is zero, this results in undefined orthonormal vectors. There are still many errors that need to be trapped, but if you stall the SVD program in addition to the code below, you should be able to solve any system, now -- provided your guesses does not result in a point $$\mathbf{x}$$ such that $$\mathbf{F}(\mathbf{x} )$$ or its Jacobian is undefined.

I will eventually write some documentation, but for now the solve is a simple Newton solver. Briefly,

$\mathbf{F}(\mathbf{x}) = \left[ \begin{array}{c} f_1 (x_1, x_2, \dotsm, x_n)\\ f_2 (x_1, x_2, \dotsm, x_n) \\ \vdots \\ f_m (x_1, x_2, \dotsm, x_n) \end{array} \right]$

where the $$f_i$$'s are the equations set to 0. For example, $$x^2 + xy = 9$$ becomes $$f(x,y) = x^2+xy - 9$$. The linearization of $$\mathbf{F}$$
is
$\mathbf{F} (\mathbf{x}_{n+1} ) = \mathbf{F} (\mathbf{x}_n ) + J_\mathbf{F} \cdot \underbrace{(\mathbf{x}_{n+1} - \mathbf{x}_{n} ) }_{\Delta \mathbf{x}}$
where $$J_{\mathbf{F}}$$ is the Jacobian (evaluated at $$\mathbf{x}_{n}$$ ). The solver solves for $$\Delta \mathbf{x}$$ by computing the pseudo inverse of $$J_\mathbf{F}$$ -- which is exactly the inverse if the Jacobian if it is non-singular. I won't get too much SVD and pseudo-inverse until I write up the documentation.

Code:
ssVersion:="Equation Library 0.063 by Han Duong"; // colors ssFG:=#0h; ssBG:=#FFFFFFh; ssR:=#FF0000h; ssG:=#FF00h; ssB:=#FFh; // error messages ssInvDat:="Invalid library data."; ssNoDat:="Unable to locate library data."; ssNullName:="Null name not allowed."; ssInvName:="Invalid name."; ssAbort:="Operation aborted; improper initialization."; ssBadVar:="Conflict with existing variable:\n"; ssSubmitBug:="This should never happen!\nPLease submit bug report."; ssNotEqn:="Non-equation on page "; ssNonVar:="No such var. on page "; ssNoLibDat:="No library data found.\nCreating empty library and empty\ninitial system."; ssEOW:="Overwrite disabled and name already exists:\n"; ssNotSaved:="SAVE CANCELED!"; ssMustChoose:="Select a system or create a new one."; ssNoVars:="No variables for this system!"; ssNoSystems:="No systems in library!"; ssBadJac:="Jacobian not invertible; singular or non-square?"; ssFUndef:="Error evaluating F!"; // app messages ssTNewVar:="Create/Edit Variable"; ssLName:="Name:"; ssLDescr:="Description:"; ssLOWrite:="Overwrite"; ssHName:="Enter the variable name"; ssHDescr:="Enter a description for the variable"; ssHOWrite:="Overwrite if variable already exists?"; ssTSaveSys:="Save Current System?"; ssHSysName:="Enter a name for the system."; ssLEqns:="Eqns:"; ssLVars:="Vars:"; ssHEqns:="Verify equations in system."; ssHVars:="Verify variables for system."; ssTNewSys:="New System"; ssTChooseSys:="Choose a System"; ssHMakeConst:="Make this variable constant?"; ssTDeleteSys:="Select Systems to Delete"; ssHMarkDel:="Mark for deletion?"; ssTSettings:="Equation Library Settings"; ssLSaving:="When switching systems, always"; ssCSaving:={ "prompt to save", "save automatically", "discard all changes"}; ssHSaving:="Select default save behavior"; // misc ssVLoadEqns:="Populating equations in Symb view..."; ssVLoadVars:="Setting up equation variables..."; ssVCheckVars:="Verifying variables in system..."; ssVCheckLibDat:="Verifying library data..."; ssVUpdateEqns:="Equations modified; updating lib data..."; ssVUpdateVars:="Variables modified; updating lib data..."; ssVNoEqns:="(No equations)"; ssVNoVars:="(No variables)"; ssVVarMade:="Created variable: "; // View menu options ssTSEP:="Select Equations Page"; ssTSelPage:="Select New Page"; ssTNewPage:="Create New Page"; ssMsgRow; export ssInit; ssLibFN; export ssCurSys; export ssCurSysIndex; export ssCurLib; ssLibSize; export ssSysTitles; ssVarKey:="SSVersion"; export ssEqPages; export ssEqPage; export ssNullSys; export ssWarning; export ssSolveVars:={}; export ssAutoVar:=1; // DO NOT MODIFY BELOW HERE ssGlobal:={ "A", "B", "C", "D", "E", "F", "G",             "H", "I", "J", "K", "L", "M", "N",             "O", "P", "R", "S", "T", "U", "V",             "W", "X", "Y", "Z", "θ",             "Z0", "Z1", "Z2", "Z3", "Z4",             "Z5", "Z6", "Z7", "Z8", "Z9" }; ssUpdateSys() begin   local n;   ssEqPages:={1,10};   n:=size(ssCurSys(2));   ssNullSys:=NOT(n);   ssEqPages(1):=ip((n-1)/10);   ssEqPages(2):=(n-1) mod 10;   ssEqPage:=0;   for n from 1 to size(ssCurSys(4)) do     if NOT(ssCurSys(7,n)) then ssSolveVars(0):=ssCurSys(4,n); end;   end;   DelAVars(AVars); end; ssInitSSApp() begin   local j,run:=1;   local cmd:="";   if ssInit then return(0); end;   dimgrob_p(G1,320,240);   ssLibFN:="Equation Library.lib";   ssVarKey:="SSVersion";   ssWarning:=1;   iferr     ssCurLib:=AFiles(ssLibFN);   then     msgbox(ssNoLibDat);     ssCurLib:={};     AFiles(ssLibFN):=ssCurLib;   end;   for j from 0 to 9 do     cmd:="E" + string(j,1,0) + ":=" + string("");     expr(cmd);   end;   // always start app with new system   ssCurSys:={ssTNewSys, {}, {}, {}, {}, {}, {} };   ssLibSize:=size(ssCurLib);   if ssLibSize then     ssSysTitles:=makelist(ssCurLib(j,1),j,1,ssLibSize);   end;   ssCurSysIndex:=0;   ssUpdateSys();   ssInit:=1; end; ssPrint(msg) begin   textout_p(msg,G1,0,12*ssMsgRow,1,ssFG,320,ssBG);   line_p(G1,0,12*ssMsgRow+12,12,12*ssMsgRow+12,ssR);   ssMsgRow:=(ssMsgRow+1) mod 20; end; ssError(msg) begin   textout_p(msg,G1,0,12*ssMsgRow,1,ssR,320,ssBG);   line_p(G1,0,12*ssMsgRow+12,12,12*ssMsgRow+12,ssR);   ssMsgRow:=(ssMsgRow+1) mod 20; end; ssWarn(msg) begin   if ssWarning then     msgbox(msg);   end; end; ssCheckType(list,t) begin   local n:=size(list);   if n then     if type(list(n))<>t then return(1); end;   else     return(1);   end;   return(0); end; //****************************************** // Library Data format // { sys1, sys2, ... , sysN } // // Each sys (system) is a list of the form // { //   "Title", //   { "eq1", "eq2", ... }, //   { sel1, sel2, ... }, //   { "var1", "var2", ... }, //   { "des1", "des2", ... }, //   { val1, val2, ... }, //   { con1, con2, ... } // } // // eq  = equation // sel = 0 or 1 for selection of corresponding //       equation in the system // var = variable // des = description of variable // val = value of corresponding variable // con = 0 or 1 for whether var is constant // // returns 0 if error or 1 if OK //****************************************** export ssCheckLibDat() begin   ssPrint(ssVCheckLibDat);   ssInitSSApp();   local syseqs,j,k;   local lib, isbad:=0;   local types:={2,2,0,2,2,0,0};   iferr     if ssLibSize then       for j from 1 to ssLibSize do         lib:=ssCurLib(j);         if type(lib)<>6 then           ssError(ssInvDat); return(0);         else           k:=size(lib);           if (k<1) OR (k>7) then             ssError(ssInvDat); return(0);           end;           for k from 2 to 7 do             if type(lib(k))<>6 then               ssError(ssInvDat); return(0);             end;             if ssCheckType(lib(k),types(k)) then               ssError(ssInvDat); return(0);             end;           end; // for k         end; // if type(lib)       end; // for j     end; // if ssLibSize   then     ssError(ssNoDat); return(0);   end;   return(1); end; // initialize variables ssInitVars() begin   local varslist:=ssCurSys(4);   local valslist:=ssCurSys(6);   local n:=size(varslist);   local j,varval;   iferr     if (AVars(ssVarKey) == ssVersion) then          return(0);     end;   then     // nop   end;   ssPrint(ssVLoadVars);   for j from 1 to n do     if pos(ssGlobal,varslist(j)) then       expr(varslist(j) + ":=" + valslist(j));     else       varval:=CAS(EVAL(varslist(j)));       purge(EVAL(varslist(j)));       if (type(CAS(EVAL(varslist(j)))) <> 8) then         ssWarn(ssBadVar+varslist(j));       end;       if ( (type(varval) == 0) OR (type(varval) == 3) ) then         AVars(varslist(j)):=varval;       else         AVars(varslist(j)):=valslist(j);       end;     end;   end;   AVars(ssVarKey):=ssVersion; end; // checks to see if AVars and system of equations // vars match; excludes version key export ssCheckVars() begin   ssPrint(ssVCheckVars);   ssInitSSApp();   ssInitVars();   local j,n;   local avarlist:=AVars();   local varlist:=ssCurSys(4);   n:=size(varlist);   for j from 1 to n do     if NOT(pos(avarlist,varlist(j)) OR pos(ssGlobal,varlist(j))) then       return(0);     end;   end;   n:=size(avarlist);   for j from 1 to n do     if ( (avarlist(j) <> ssVarKey) AND NOT(pos(varlist,avarlist(j))) ) then       return(0);     end;   end;   return(1); end; // do we have any equations //  1: yes, equations in symb view //  0: no equations at all // -1: yes, but only saved in library ssHaveEqns() begin   local j;   for j from 0 to 9 do     if ISCHECK(j) then return(1); end;   end;   if (ssLibSize == 0) then return(0); end;   if (type(ssCurSys) <> 6) then return(0); end;   if size(ssCurSys) then     if size(ssCurSys(2)) then       return(-1);     else       return(0);     end;   else     return(0);   end; end; // UI for creating new variables ssCreateVar() begin   local run:=1;   local n;   local varname:="", varinfo:="";   local ow:=0;   ssInitSSApp();   ssInitVars();   while run do     if input(       {         {varname, [2], {30, 65, 0}},         {varinfo, [2], {30, 65, 1}},         {ow, 1, {30,10,2}}       },       ssTNewVar,       { ssLName, ssLDescr, ssLOWrite },       { ssHName, ssHDescr, ssHOWrite }     )     then       if size(varname)<1 then         msgbox(ssNullName);       else         n:=pos(AVars,varname);         if n then           if ow then             n:=pos(ssCurSys(4),varname);             ssCurSys(5,n):=varinfo;             run:=0;           else             msgbox(ssEOW + varname);           end;         else           iferr             if NOT(pos(ssGlobal,varname)) then               AVars(varname):=0;             end;           then             msgbox(ssInvName);           else             purge(EVAL(varname));             ssCurSys(4,0):=varname;             ssCurSys(5,0):=varinfo;             ssCurSys(6,0):=0;             ssCurSys(7,0):=0;             msgbox(ssVVarMade + varname);             varname:=""; varinfo:="";           end;         end;       end; // end null name     else       run:=0;     end;   end; // end while end; // save current equations ssSaveEqns() begin   local j,k,n1,n2;   local cmd:="";   local oldeqn:="";   local neweqn:="";   local markadd:=0;   local neweqns:={};   local teqnlist:={};   local tsellist:={};   if ssCheckVars() then     ssPrint(ssVLoadEqns);     n1:=ssEqPage*10+1;     if (ssEqPage < ssEqPages(1)) then       n2:=n1+9;     else       n2:=n1+ssEqPages(2);     end;     for j from n1 to n2 do       k:= j mod 10;       cmd:="string(E" + string(k,1,0) + ",1,12)";       iferr         neweqn:=expr(cmd);       then         neweqn:="";       else         if ssNullSys then           oldeqn:="";         else           cmd:="E" + string(k,1,0) + ":=" + string(ssCurSys(2,j));           iferr expr(cmd); then end;           cmd:="string(E" + string(k,1,0) + ",1,12)";           iferr oldeqn:=expr(cmd); then end;         end;         if (oldeqn <> neweqn) then           if ssNullSys then             ssCurSys(2,0):=neweqn;           else             ssCurSys(2,j):=neweqn;           end;         end;       end;       cmd:="E" + string(k,1,0) + ":=" + string(neweqn);       iferr expr(cmd); then         // msgbox(ssSubmitBug);       end;       if ISCHECK(k) then         ssCurSys(3,j):=1;       else         ssCurSys(3,j):=0;       end;       if (size(neweqn) == 0) then         ssCurSys(3,j):=-1; // mark for delete       end;     end;  // end for j     n2:=n1+9;     for k from j to n2 do       n1:=k mod 10;       cmd:="string(E" + string(n1,1,0) + ",1,12)";       iferr         neweqn:=expr(cmd);       then         // nop       else         if size(neweqn) then           neweqns(0):=neweqn;           neweqns(0):=ISCHECK(n1);           markadd:=1;         end;       end;     end; // end for k     // rebuilt existing equations     k:=0;     n2:=size(ssCurSys(2));     for j from 1 to n2 do       if (ssCurSys(3,j) >= 0) then         teqnlist(0):=ssCurSys(2,j);         tsellist(0):=ssCurSys(3,j);         k:=k+1;       end;     end;     ssCurSys(2):=teqnlist;     ssCurSys(3):=tsellist;     if markadd then       n2:=size(neweqns)/2;       for j from 1 to n2 do         ssCurSys(2,0):=neweqns(2*j-1);         ssCurSys(3,0):=neweqns(2*j);         k:=k+1;       end;     end;     ssEqPages(1):=ip((k-1)/10);     ssEqPages(2):=(k-1) mod 10;     ssEqPage:=min(max(0,ssEqPage),ssEqPages(1));     if k then       ssNullSys:=0;     else       ssNullSys:=1;     end;     return(1);   else     msgbox(ssAbort);     return(0);   end; end; // set up equations in Symb view export ssLoadSymb() begin   local j,k,n1,n2;   local cmd:="";   // no equations to populate Symb view?   if ssNullSys then      for j from 0 to 9 do       cmd:="E" + string(j,1,0) + ":=" + string("");       expr(cmd);     end;     return(1);   end;   ssPrint(ssVLoadEqns);   n1:=ssEqPage*10+1;   if (ssEqPage < ssEqPages(1)) then     n2:=n1+9;   else     n2:=n1+ssEqPages(2);   end;   for j from n1 to n2 do     k:= j mod 10;     cmd:="E" + string(k,1,0) + ":=" + string(ssCurSys(2,j));     iferr expr(cmd); then end;     if (ssCurSys(3,j)==1) then CHECK(k); else UNCHECK(k); end;   end;   n2:=n1+9;   for k from j to n2 do     n1:=k mod 10;     cmd:="E" + string(n1,1,0) + ":=" + string("");     expr(cmd);   end;   return(1); end; // select an equations page ssSelectEqPage() begin   ssInitSSApp();   local j;   local n:=ssEqPages(1)+1;   local page:=ssEqPage+1;   local curpage:=page;   local pagelist:={};   local title:="[" + string(page,1,0) + "/" + string(ssEqPages(1)+1,1,0) + "] " + ssTSelPage;      pagelist:=makelist(j,j,1,n);   pagelist(0):=ssTNewPage;   if choose(page,title,pagelist) then     if (curpage <> page) then       ssSaveEqns();        if (page == (n+1)) then         if (ssNullSys == 0) then            ssEqPages(1):=n+1;           // fill last page           if (ssEqPages(2) < 9) then             n:=9-ssEqPages(2);             for j from 1 to n do               ssCurSys(2,0):="";               ssCurSys(3,0):=-1; // mark for deletion             end;           end;         end;         // create new page         for j from 1 to 10 do           ssCurSys(2,0):="";           ssCurSys(3,0):=-1; // mark for deletion         end;         n:=size(ssCurSys(2));         ssEqPage:=ip((n-1)/10);         ssEqPages(1):=ssEqPage;         ssEqPages(2):=(n-1) mod 10;         ssNullSys:=0;       else         // not creating new blank page         ssEqPage:=min(max(0,page-1),ssEqPages(1));       end; // if newpage or existing page     end; // if selpage <> page     ssLoadSymb();   end; // if choose end; // set new system export ssSetSystem(n) begin   if (n > ssLibSize) then     return(0);   end;   ssCurSysIndex:=n;   ssCurSys:=ssCurLib(n);   ssUpdateSys();   ssInitVars();   ssLoadSymb(); end; ssErrorPage(n,msg,eqn) begin   local p:=ip((n-1)/10)+1;   msgbox(msg + n + ":\n" + eqn); end; // automatically add existing variable to vars list // should only happen with global variables ssAutoAddVar(var) begin   local val;   local t;   iferr     val:=expr(var);   then     val:=0;   end;   t:=type(val);   if ( (t<>0) OR (t<>3) ) then     val:=0;   end;      // convert existing vars to app var   // exceptions: global system vars cannot   // be app vars in current firmware   if NOT(pos(ssGlobal,var)) then     AVars(var):=val;   end;   ssCurSys(4,0):=var;   ssCurSys(5,0):="";   ssCurSys(6,0):=val;   ssCurSys(7,0):=0; end; // check if equations are valid // assumes equations have been saved export ssCheckEqns() begin   local n:=size(ssCurSys(2));   local eqn:="";   local j,k,v;   local varlist:={};   local var:="";   if n then     // check each equation     for j from 1 to n do       eqn:=ssCurSys(2,j);       // do we have = symbol?       if instring(eqn,"=") then         iferr eqn:=string(E0,1,12); then eqn:=""; end;         E0:=ssCurSys(2,j);         varlist:=LNAME(E0);         v:=size(varlist);         if v then           for k from 1 to v do             var:=string(varlist(k));             if (pos(ssCurSys(4),var) == 0) then               if ssAutoVar then                 ssAutoAddVar(var);               else                 ssErrorPage(ip((j-1)/10)+1,ssNonVar,ssCurSys(2,j)+"\n> "+varlist(k));                 iferr E0:=eqn; then end;                 return(0);               end;             end;           end;         else           // constants only? not equation           ssErrorPage(j,ssNotEqn,ssCurSys(2,j));           iferr E0:=eqn; then end;           return(0);                end;         iferr E0:=eqn; then end;                else         // no equal symbol; report page         ssErrorPage(j,ssNotEqn,eqn);         return(0);       end;     end;     // all eqns seem ok     return(1);   else     // no equations to check     return(-1);   end; end; // save current system // no null names but does not check // validity of names beyond whether // AVars will accept such a name ssSaveSystem() begin   ssSaveEqns();   local sysname:=ssCurSys(1);   local eqns:=ssCurSys(2);   local vars:=ssCurSys(4);   local eqn,var,n;   local run:=1;   // bug in input when choose lists are empty   if (size(eqns) < 1) then eqns:={ ssVNoEqns }; end;   if (size(vars) < 1) then vars:={ ssVNoVars }; end;   while run do     if input(       {         { sysname, [2], { 15, 75, 0 } },         { eqn, eqns, { 15, 75, 1 } },         { var, vars, { 15, 75, 2 } }       },       ssTSaveSys,       { ssLName,    ssLEqns, ssLVars },       { ssHSysName, ssHEqns, ssHVars }     )     then       if size(sysname) then         // maybe later add check for dupes         n:=pos(ssSysTitles, sysname);         if ssCheckEqns() then           ssCurSys(1):=sysname;           ssCurLib(ssCurSysIndex):=ssCurSys;           AFiles(ssLibFN):=ssCurLib;           if (ssCurSysIndex == 0) then ssCurSysIndex:=size(ssCurLib); end;           ssLibSize:=size(ssCurLib);         else           ssWarn(ssNotSaved);         end;         run:=0;       else         msgbox(ssNullName);       end;     else       // user canceled       ssWarn(ssNotSaved);       run:=0;     end;   end; // end while; end; // create a system ssNewSystem() begin   ssCurSys:={ssTNewSys, {}, {}, {}, {}, {}, {} };   ssCurSysIndex:=0;   ssUpdateSys();   ssInitVars();   ssLoadSymb();   startview(0,1); end; ssChooseSys() begin   local n,t;   local run:=1;   local systitles:={};   n:=ssCurSysIndex;   iferr     ssCurLib:=AFiles(ssLibFN);   then     ssWarn(ssNoLibDat);     ssCurLib:={};     AFiles(ssLibFN):=ssCurLib;   end;   ssLibSize:=size(ssCurLib);   if ssLibSize then     systitles:=makelist(ssCurLib(n,1),n,1,ssLibSize);   else     systitles:={};   end;   ssSysTitles:=systitles;   systitles(0):=ssTNewSys;   t:=choose(n, ssTChooseSys, systitles);   if t then     if ( n <= ssLibSize ) then       ssSetSystem(n);     else       ssNewSystem();     end;     return(1);   end;   return(0); end; // UI for deleting systems export ssDeleteSys() begin   local cmd:="input({";   local systitles;   local j;   local selected:={};   local lib:={};   if ssLibSize then     selected:=makelist(0,j,1,ssLibSize);     systitles:=makelist(ssCurLib(j,1),j,1,ssLibSize);     for j from 1 to ssLibSize do       cmd:=cmd + "{selected(" + string(j,1,0) + "),0,{94,5," + string(j-1,1,0) + "}}";       if (j<ssLibSize) then cmd:=cmd + ",\n"; end;     end;     cmd:=cmd + "},\n" + string(ssTDeleteSys) + ",\n{";     for j from 1 to ssLibSize do       cmd:=cmd + string(ssCurLib(j,1));       if (j<ssLibSize) then cmd:=cmd + ",\n"; end;     end;     cmd:=cmd + "},\n{";     for j from 1 to ssLibSize do       cmd:=cmd + string(ssHMarkDel);       if (j<ssLibSize) then cmd:=cmd + ",\n"; end;     end;     cmd:=cmd + "},\n" + string(selected,1,0) + ",\n" + string(selected,1,0) + ")";     Notes("eqlib_debug.log"):=cmd;     j:=expr(cmd);     if j then       // current system deleted? if so set index to 0       if ssCurSysIndex then         if selected(ssCurSysIndex) then           ssCurSysIndex:=0;         end;       end;       // rebuild library and adjust index       for j from 1 to ssLibSize do         if NOT(selected(j)) then           lib(0):=ssCurLib(j);           if (j < ssCurSysIndex) then             ssCurSysIndex:= ssCurSysIndex - 1;           end;         end;       end;       ssCurLib:=lib;       AFiles(ssLibFN):=ssCurLib;     end;   else     ssWarn(ssNoSystems);   end; end; // variable browser export ssVarBrowser() begin   local vals:=ssCurSys(6);   local cons:=ssCurSys(7);   local cmd:="input({";   local n:=size(vals);   local j,k;   local tmp:="";   local xloc:="15,75";   if (n<1) then     msgbox(ssNoVars);     return(0);   end;   k:=1;   for j from 1 to n do     k:=max(k,size(ssCurSys(4,j)));   end;   if (k > 3) then xloc:="30,60"; end;   // build input( ... )   // { variable, type, { pos } }   for j from 1 to n do     k:=pos(ssGlobal, ssCurSys(4,j));     if (k<28) then tmp:="0"; else tmp:="0,3"; end;          cmd:=cmd + "{vals(" + string(j,1,0) + "),[" + tmp + "],{" + xloc + "," + string(j-1,1,0) + "}}, ";     cmd:=cmd + "{cons(" + string(j,1,0) + "),0,{94,5," + string(j-1,1,0) + "}}";     if ((n>1) AND (j<n)) then cmd:=cmd + ",\n"; end;   end;   // add title   cmd:=cmd + "},\n" + string(ssCurSys(1) + " Variables") + ",\n{";   // add labels   for j from 1 to n do     cmd:=cmd + string(ssCurSys(4,j) + "=") + ", " + string("");     if ((n>1) AND (j<n)) then cmd:=cmd + ",\n"; end;   end;   cmd:=cmd + "},\n{";   // add help   for j from 1 to n do     cmd:=cmd + string(ssCurSys(4,j) + ": " + ssCurSys(5,j)) + "," + string(ssHMakeConst);     if ((n>1) AND (j<n)) then cmd:=cmd + ",\n"; end;   end;   cmd:=cmd + "},\n";   // add init and reset vals   tmp:="{";   for j from 1 to n do     tmp:=tmp + vals(j) + "," + cons(j);     if ((n>1) AND (j<n)) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "}";   cmd:=cmd + tmp + ",\n" + tmp + ")";   Notes("eqlib_debug.log"):=cmd;   j:=expr(cmd);   if j then     ssCurSys(6):=vals;     ssCurSys(7):=cons;     for j from 1 to n do       if pos(ssGlobal,ssCurSys(4,j)) then         expr(ssCurSys(4,j) + ":=" + vals(j));       else         AVars(ssCurSys(4,j)):=vals(j);       end;     end;   end;   return(j); end; // build F, F(var1, var2, ...) and Jacobian // assumes all equations and variables have been // validated; also assumes equations are saved and // that selectd equations form a non-null list // saves F as ssF (CAS variable) export casMakeF() begin   local cmd:="['";   local cmd2:="[";   local j,n;   local varlist:={};   local eqnlist:={};   local k:=size(ssCurSys(4));   local vstr1:="";   local vstr2:="";   n:=size(ssCurSys(2));   for j from 1 to n do     if ssCurSys(3,j) then       eqnlist(0):=ssCurSys(2,j);     end;   end;   for j from 1 to k do     if NOT(ssCurSys(7,j)) then       varlist(0):=j;     end;   end;   ssSolveVars:=varlist;   n:=k-1;   for j from 1 to n do     vstr1:=vstr1 + ssCurSys(4,j) + ",";   end;   vstr1:=vstr1 + ssCurSys(4,k);   n:=size(varlist)-1;   for j from 1 to n do     vstr2:=vstr2 + ssCurSys(4,varlist(j)) + ",";   end;   vstr2:=vstr2 + ssCurSys(4,varlist(n+1));   // ssF:=( vars ) -> [ f1, f2, ... ]   n:=size(eqnlist)-1;   for j from 1 to n do     cmd:=cmd + eqnlist(j) + ")','";     cmd2:=cmd2 + eqnlist(j) + "),";   end;   cmd:=cmd + eqnlist(n+1) + ")']";   cmd:=replace(cmd,"=","-(");   CAS(EVAL("ssF:=" + cmd));   cmd2:=cmd2 + eqnlist(n+1) + ")]";   cmd2:=replace(cmd2,"=","-(");   CAS(EVAL("ssFx:=(" + vstr1 + ")->(" + cmd2 + ")"));   cmd:="ssJ:=transpose(diff(" + cmd + ",[" + vstr2 + "]))";   CAS(EVAL(cmd)); end; // pseudo inverse using SVD export ssPInv() begin   local l:=CAS("svd(ssJ)");   local r:=size(l(2)); r:=r(1);   local j,n,ns,s;   local tol:=1E-30;   local d:=[[0]];   // built-in SVD fails on some singular matrices   // failed SVD matrices returned as list of lists   if ((type(l(1)) == 6) OR (type(l(3)) == 6)) then     d:=CAS("pinv(ssJ,1e-30)");     return(d);   end;   // sort sing. vals from largest to smallest   // and adjust col/row accordingly   for n from r downto 2 do     ns:=1;     s:=l(2,1);     for j from 2 to n do       if (l(2,j) <= s) then         ns:=j; s:=l(2,j);       end;     end;     if (ns < n) then       l(2,ns):=l(2,n);       l(2,n):=s;       d:=l(1);       l(1):=swapcol(d,ns,n);       d:=l(3);       l(3):=swapcol(d,ns,n);     end;   end;   for n from 1 to r do     s:=l(2,n);     if (s >= tol) then       l(2,n):=1/s;     else       l(2,n):=0;     end;   end;   j:=size(l(3)); n:=j(1);   j:=size(l(1)); r:=j(1);   d:=makemat(ifte(I==J,l(2,I),0),n,r);   // make sure to use CAS.trn for Hermetian   return(l(3)*d*CAS.trn(EVAL(l(1)))); end; // Newton's Method // dx = - J^(-1) * F // x[n+1] = x[n] + dx export casNewton() begin   local j,k,nsv,nv;   local var;   local tolx:=1.0E-8;   local tolF:=1.0E-10; // exit if F.F < tolF   local nsF=2*tolF; // initial squared norm of F   local vecNewt:={}; //  L1:={}; // for debugging only   local pinv:=[[0]];   local dx:=2*tolx;   nv:=size(ssCurSys(4));   nsv:=size(ssSolveVars);   for j from 1 to nsv do     vecNewt(0):=ssCurSys(6,ssSolveVars(j)); //    L1(0):=ssCurSys(6,ssSolveVars(j));   end;      for k from 1 to 100 do     nsF:=CAS("approx(sqrt(ssF*ssF))");     if (type(nsF) == 2) then       msgbox(ssFUndef);       break;     end;     vecNewt(0):=nsF;     if (k == 1) then        vecNewt(0):=0;     else       vecNewt(0):=abs(dx);     end;     if (nsF < tolF) then break; end;     if (abs(dx) < tolx) then break; end;     iferr       pinv:=ssPInv();       dx:=-pinv*CAS.transpose("approx(ssF)");     then       msgbox(ssBadJac);       break;     end;     for j from 1 to nsv do       var:=ssCurSys(4,ssSolveVars(j));       if pos(ssGlobal,var) then         expr(var + ":=" + var + "+" + dx(j,1));       else         AVars(var):=AVars(var) + dx(j,1);       end;       ssCurSys(6,ssSolveVars(j)):=expr(var);     end; // for j     for j from 1 to nsv do       vecNewt(0):=ssCurSys(6,ssSolveVars(j)); //      L1(0):=ssCurSys(6,ssSolveVars(j));     end;   end; // for k   return(CAS.list2mat(EVAL(vecNewt),EVAL(nsv+2))); end; START() begin //  startview(-1,1);   ssInitSSApp();   ssInitVars();   startview(0,1); end; // Equations Editor Symb() begin   if ssInit then     ssSaveEqns();   else     ssInitSSApp();     ssInitVars();   end;   ssLoadSymb();   startview(0,1); end; // System of Equations browswer Plot() begin   if ssInit then     ssSaveSystem();       else     ssInitSSApp();     ssInitVars();   end;   ssChooseSys();   startview(0,1); end; // run solver Num() begin   local matNewt:=[[0]];   local j;   local tlist:={};   local labels:={};   if ssInit then     ssSaveEqns();     if NOT(ssCheckEqns()) then       return(0);     end;     if NOT(ssHaveEqns()) then       return(0); // exit     end;     if ssVarBrowser() then       casMakeF();       matNewt:=casNewton();       tlist(0):=ssCurSys(1);       tlist(0):={};       for j from 1 to size(ssSolveVars) do         labels(0):=ssCurSys(4,ssSolveVars(j));       end;       labels(0):="ǁFǁ";       labels(0):="ǁΔxǁ";       tlist(0):=labels;       editmat(matNewt,tlist);     end;   else     ssInitSSApp();     ssInitVars();   end; end; // Set up initial guess  NumSetup() begin   if ssInit then ssSaveEqns(); end;   ssVarBrowser(); end; // select equations page (View Menu) view "Select Page", ssMenuEqnPage() begin   ssSelectEqPage();   startview(0,1); end; // UI for creating new variables (View Menu) view "Add/Edit Variable", ssMenuNewVar() begin   ssCreateVar(); end;  // reload saved equations (View Menu) view "Restore Equations", ssMenuLoadEqns() begin   if ssCurSysIndex then     ssCurSys:=ssCurLib(ssCurSysIndex);     ssUpdateSys();     ssInitVars();   end;   ssLoadSymb();   startview(0,1); end; // save current system (View Menu) view "Save System", ssMenuSaveSys() begin   ssSaveSystem(); end; // create new system (View Menu) view "New System", ssMenuNewSys() begin   ssSaveSystem();   ssNewSystem(); end; // delete existing system (View Menu) view "Delete System", ssMenuDelSys() begin   ssSaveSystem();   ssDeleteSys(); end; // select system (View Menu) view "Select System", ssMenuSetSys() begin   ssSaveSystem();   if ssChooseSys() then startview(0,1); end; end;

Original first post below -- info is out of date due to revisions
-------------------------------------------------------------------------------------------
Here's a brief outline of my ideas.

1) Each category of equations will be a list of lists of the following form:

Code:
{   "Title",   { "equation1", "equation2", ..., "equationn" },   { "var1", "var2", ..., "varn" },   { "vardescription1", "vardescription2", ..., "vardescriptionn" } }

Perhaps later on we can include graphics by means of lists of point definitions, line definitions, and polygon definitions. Ideally, the user would not have to specify what the variable list is, and the app will be smart enough to parse through each equation string to identify what the variables are.

2) A choose screen will allow users to select which category of equations to work with

3) Once a category is selected, all the equations will be shown in an INPUT() window with check boxes next to them. The user then checks the equations to use.

4) The app then creates another INPUT() window for variables and their initial values. Each variable may also be checked to specify whether the quantity should be treated as a variable that may change during the solve process, or if it is a constant (much like in the SOLVESYS library for the HP48 and HP49).

5) The equations are then modified to insert constants where appropriate, and then call fsolve() to solve the system.

This is the general scheme. What would also be nice is an interface for users to add in new categories of equations. Anyway, this is something for a later update. For now, I'll start working on building the initial interface with the assumption that categories of equations already exist. This is where, if you want to contribute, you can help by posting the categories as shown in the format above.

Code:
{   "Linear Motion",   {      "x=x0+v0*t+1/2*a*t^2",     "x=x0+v*t-1/2*a*t^2",     "x=x0+1/2*(v0+v)*t",     "v=v0+a*t"   },   { "x", "x0", "v0", "v", "t", "a" },   {     "final position",     "initial position",     "initial velocity",     "final velocity",     "time",     "acceleration"   } }

Graph 3D | QPI | SolveSys
01-09-2015, 09:56 PM
Post: #2
 ArielPalazzesi Member Posts: 90 Joined: Dec 2013
RE: Creating an equation library
Hi Han!

I like the way you've addressed the development of the library. I'll see your progress closely .

A hug .
01-10-2015, 03:45 AM (This post was last modified: 01-10-2015 04:07 AM by Han.)
Post: #3
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library
Here's a very basic skeleton. This version does not handle systems that have free variables. Ideally, in the case of free variables, then the initial guesses should be used to establish a solution. For example, if (x,y,z) = (x,2x-1,3x+1), then the initial guess of (4,5,6) should produce the solution (4,7,13) using x=4. This will be implemented over time. For now, what you get is basically a proof of concept.

Code:
export eqldata; export varlist; export eqlist; export eql_results; export inputstr; eql_init(); eql_cmenu(); eql_setup(); eql_dovars(); eql_dosolve(); export eqlib() begin   eql_init();   local c:=eql_cmenu();   local tmp;   if c then     tmp:=eql_setup(c);   end;   if tmp then     tmp:=eql_dovars(c,1);   end;   if tmp then     repeat       eql_dosolve(c);       tmp:=eql_dovars(c,0);     until tmp==0;   end; end; eql_init() begin   local tmp;   iferr tmp:=Notes("EqLib") then     msgbox("No equation library data!");   else     eqldata:=expr(tmp);   end; end; eql_cmenu() begin   local c;   local n:=size(eqldata);   local categories:=makelist(eqldata(X,1),X,1,n);   choose(c,"Equation Library", categories);   return(c); end; eql_setup(c) begin   local eqns:=eqldata(c);   local tmp:="input({";   local n:=size(eqns(2));   local i;   eqlist:=makelist(1,X,1,n);   for i from 1 to n do     tmp:=tmp+"{eqlist(" + string(i,1,12) + "),0,{94,5," + string(i-1,1,12) + "}}";     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "}," + string(eqns(1)) + ",{";   for i from 1 to n do     tmp:=tmp + string(eqns(2,i));     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},{";   for i from 1 to n do     tmp:=tmp + string("Use this equation?");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "})";   n:=expr(tmp);   return(n); end; eql_dovars(c,f) begin   local eqns:=eqldata(c);   local tmp:="";   local n:=size(eqns(3));   local i; local t;   if f then     varlist:=makelist(0,X,1,n);   end;   for i from 1 to n do     tmp:="type(" + eqns(3,i) + ")";     if CAS(tmp)==6 then tmp:=eqns(3,i) + ":=0"; CAS(tmp); end;   end;   tmp:="input({";   for i from 1 to n do     tmp:=tmp + "{" + eqns(3,i) + ",[0],{30,30," + string(i-1,1,12) + "}},";     tmp:=tmp + "{varlist(" + string(i,1,12) + "),0,{94,5," + string(i-1,1,12) + "}}";     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "}," + string(eqns(1) + " Variables") + ",{";   for i from 1 to n do     tmp:=tmp + string(eqns(3,i) + "=") + "," + string("");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},{";   for i from 1 to n do     tmp:=tmp + string("Enter the value for the " + eqns(4,i)) + ",";     tmp:=tmp + string("Make this variable a constant?");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},";   local tmp2:="{";   for i from 1 to n do     tmp2:=tmp2 + CAS(eqns(3,i)) + "," + varlist(i);     if (n>1) AND (i<n) then tmp2:=tmp2 + ","; end;   end;   tmp2:=tmp2 + "}";   tmp:=tmp + tmp2 + "," + tmp2 + ")";   t:=expr(tmp);   if t then     for i from 1 to n do       if varlist(i)==0 then         tmp:="purge(" + eqns(3,i) + ")";         CAS(tmp);       end;     end;   end;      return(t); end; eql_dosolve(c) begin   local eqns:=eqldata(c);   local n:=size(eqns(2));   local i;   local tmp:="fsolve([";   local k:=ΣLIST(eqlist);   for i from 1 to n do     if eqlist(i) then       k:=k-1;       tmp:=tmp + eqns(2,i);       if k then tmp:=tmp + ","; end;     end;   end;   tmp:=tmp + "],[";   n:=size(eqns(3));   k:=n-ΣLIST(varlist);   for i from 1 to n do     if varlist(i)==0 then       k:=k-1;       tmp:=tmp + eqns(3,i);       if k then tmp:=tmp + ","; end;     end;   end;   tmp:="eql_results:=" + tmp + "])";   inputstr:=tmp;   CAS(tmp);   local l:=size(eql_results);   if type(eql_results)==6 then msgbox("No solution or infinite solutions; not implemented"); end; // if multiple solutions; just get the first one; fix this later to choose "best" solution   if type(eql_results)==4 then     k:=1;     for i from 1 to n do       if varlist(i)==0 then         tmp:=eqns(3,i) + ":=" + eql_results(1,k);         CAS(tmp);         k:=k+1;       end;     end;   end; end;

You must first create a Note named "EqLib" whose contents is:
Code:
{   {     "Linear Motion",     {        "x=x0+v0*t+1/2*a*t^2",       "x=x0+v*t-1/2*a*t^2",       "x=x0+1/2*(v0+v)*t",       "v=v0+a*t"     },     { "x", "x0", "v0", "v", "t", "a" },     {       "final position",       "initial position",       "initial velocity",       "final velocity",       "time",       "acceleration"     }   } }

Run eqlib() and choose "Linear Motion" (the only option). The next screen allows you to pick which equations to use. Leave all selected. The screen after that allows you to set up the known values. Those which should be treated as constants can be checked so that they are not considered a variable in the system. Press OK. If a solution exists, the values are updated. Press cancel to quit. I chose x=1, x0=5, t=3, and a=9.8. This results in v0=-16.033333... and v=13.366666...

Again, this is just a proof of concept. There are still a number of things I have to figure out about fsolve. Even when a system is solvable, if we give fsolve initial conditions, we sometimes get "Invalid dimension" errors. It appears that if fsolve is given more equations than the number of variables, it claims invalid dimension even though that may not be the case (e.g. some of the equations are equivalent).

An alternative to fsolve would be to implement our own system solver using something similar to that in SOLVESYS for the HP48/HP49. A brief glance at the source code suggests a Newton-like algorithm using the Jacobian.

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01-10-2015, 04:12 PM
Post: #4
 Giancarlo Member Posts: 256 Joined: Dec 2013
RE: Creating an equation library (updated)
Hello Han,
thank you so much for working on an equation library. I tried to use it in the EMU. Everything went fine but i receive the "not supported" msgbox for any value assigned to the variables.

Apart from this I would like to understand why you chose to store the equations in a note and not in a simple list. Which advantages you see in doing like this?

Second question, why do you prefer to use fsolve() instead of programming the solve app? just to manage the overall process?

Thanks

Giancarlo
01-10-2015, 04:34 PM (This post was last modified: 01-10-2015 04:43 PM by Han.)
Post: #5
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
(01-10-2015 04:12 PM)Giancarlo Wrote:  Hello Han,
thank you so much for working on an equation library. I tried to use it in the EMU. Everything went fine but i receive the "not supported" msgbox for any value assigned to the variables.

Apart from this I would like to understand why you chose to store the equations in a note and not in a simple list. Which advantages you see in doing like this?

Second question, why do you prefer to use fsolve() instead of programming the solve app? just to manage the overall process?

Thanks

Giancarlo

If the system and initial guesses result in anything other than a single solution, then the generic "not supported" message will appear. The current skeleton of an engine does not handle any case beyond the single-solution case (for the moment; that is not the plan for the future, of course). If you simply entered initial guesses without specifying any of the "variables" to be constants, then there would be free variables (since we have more variables than equations), resulting in a system with infinite solutions.

As for why I stored the data in notes -- it's to prevent the erasure of data when one opens the source code of the program. Any variable associated with the program will be reset. Now, we could embed the data into the source code, but as you know, pressing [Shift][Esc] erases the entire program without any prompts. I don't know how many times I have cursed the virtual calculator or the real calculator because I didn't notice that the [Shift] operator was already active and accidentally erased my entire program. Saving it in the notes means we avoid this issue. Since the plan is to also implement an interface to add equations and manage the library, there is no real reason for any user to manually edit the corresponding notes (since it too has the same issue as programs with the [Shift][Esc]).

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01-10-2015, 06:39 PM (This post was last modified: 01-10-2015 06:50 PM by akmon.)
Post: #6
 akmon Member Posts: 197 Joined: Jun 2014
RE: Creating an equation library (updated)
Hello Han. Your Project is very interesting, and believe me, I was thinking of programming something similar to Solvesys. In my case using a process to extract the variables of the equations, avoiding to make a variable list, excepting if you deliberately insert a list explaining its purpose, like your example. So it´s a good idea to extract them automatically, as you wrote. Don´t know if there is a command like TVARS on HP 49 to get the variables of a formula.

By the way, I cannot find the solutions on you examples in my emulator. After inserting values a Box appears saying: no solutions or infinite solutions, and then time value returns to 0. I´ll try on my physical calculator.

Another suggestion, the possibility to go back to the formula menu from the variable menu.

And finally, fsolve Works better than solve App, let´s see if it behaves well with more examples. With this update (6975), I managed to solve a system of equations using fsolve where solve App couldn´t.
01-10-2015, 08:41 PM (This post was last modified: 01-10-2015 08:56 PM by Han.)
Post: #7
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
To obtain the answers I got, you must specify that x, x0, t, and a are constants (check box) next to the variable.

Edit: I see the issue; there is a priority conflict between local and CAS variables. This should be fixed relatively soon.

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01-10-2015, 08:49 PM
Post: #8
 akmon Member Posts: 197 Joined: Jun 2014
RE: Creating an equation library (updated)
I did it, but no luck.

Attached File(s) Thumbnail(s)

01-10-2015, 09:43 PM (This post was last modified: 01-10-2015 09:44 PM by Han.)
Post: #9
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
Try this slight update:
Code:
export eql_data; export eql_varlist; export eql_eqnslist; export eql_initvals; export eql_results; // this is for debugging export inputstr; eql_init(); eql_cmenu(); eql_setup(); eql_dovars(); eql_dosolve(); export eqlib() begin   eql_init();   local c:=eql_cmenu();   local tmp;   if c then     tmp:=eql_setup(c);   end;   if tmp then     tmp:=eql_dovars(c,1);   end;   if tmp then     repeat       eql_dosolve(c);       tmp:=eql_dovars(c,0);     until tmp==0;   end; end; eql_init() begin   local tmp;   iferr tmp:=Notes("EqLib") then     msgbox("No equation library data!");   else     eql_data:=expr(tmp);   end; end; eql_cmenu() begin   local c;   local n:=size(eql_data);   local categories:=makelist(eql_data(X,1),X,1,n);   choose(c,"Equation Library", categories);   return(c); end; eql_setup(c) begin   local eqns:=eql_data(c);   local tmp:="input({";   local n:=size(eqns(2));   local i;   eql_eqnslist:=makelist(1,X,1,n);   for i from 1 to n do     tmp:=tmp+"{eql_eqnslist(" + string(i,1,12) + "),0,{94,5," + string(i-1,1,12) + "}}";     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "}," + string(eqns(1)) + ",{";   for i from 1 to n do     tmp:=tmp + string(eqns(2,i));     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},{";   for i from 1 to n do     tmp:=tmp + string("Use this equation?");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "})";   n:=expr(tmp);   return(n); end; eql_dovars(c,f) begin   local eqns:=eql_data(c);   local tmp:="";   local n:=size(eqns(3));   local i; local t;   if f then     eql_varlist:=makelist(0,X,1,n);     eql_initvals:=makelist(0,X,1,n);   end;   tmp:="input({";   for i from 1 to n do     tmp:=tmp + "{eql_initvals(" + string(i,1,12) + "),[0],{30,30," + string(i-1,1,12) + "}},";     tmp:=tmp + "{eql_varlist(" + string(i,1,12) + "),0,{94,5," + string(i-1,1,12) + "}}";     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "}," + string(eqns(1) + " Variables") + ",{";   for i from 1 to n do     tmp:=tmp + string(eqns(3,i) + "=") + "," + string("");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},{";   for i from 1 to n do     tmp:=tmp + string("Enter the value for the " + eqns(4,i)) + ",";     tmp:=tmp + string("Make this variable a constant?");     if (n>1) AND (i<n) then tmp:=tmp + ","; end;   end;   tmp:=tmp + "},";   local tmp2:="{";   for i from 1 to n do     tmp2:=tmp2 + eql_initvals(i) + "," + eql_varlist(i);     if (n>1) AND (i<n) then tmp2:=tmp2 + ","; end;   end;   tmp2:=tmp2 + "}";   tmp:=tmp + tmp2 + "," + tmp2 + ")";   t:=expr(tmp);   if t then     for i from 1 to n do       if eql_varlist(i) then         tmp:=eqns(3,i) + ":=" + eql_initvals(i);       else         tmp:="purge(" + eqns(3,i) + ")";       end;       CAS(tmp);     end;   end;      return(t); end; eql_dosolve(c) begin   local eqns:=eql_data(c);   local n:=size(eqns(2));   local i;   local tmp:="fsolve([";   local k:=ΣLIST(eql_eqnslist);   for i from 1 to n do     if eql_eqnslist(i) then       k:=k-1;       tmp:=tmp + eqns(2,i);       if k then tmp:=tmp + ","; end;     end;   end;   tmp:=tmp + "],[";   n:=size(eqns(3));   k:=n-ΣLIST(eql_varlist);   for i from 1 to n do     if eql_varlist(i)==0 then       k:=k-1;       tmp:=tmp + eqns(3,i);       if k then tmp:=tmp + ","; end;     end;   end;   tmp:="eql_results:=" + tmp + "])";   inputstr:=tmp;   CAS(tmp);   local l:=size(eql_results);   if type(eql_results)==6 then msgbox("No solution or infinite solutions; not implemented"); end; // if multiple solutions; just get the first one; fix this later to choose "best" solution   if type(eql_results)==4 then     k:=1;     for i from 1 to n do       if eql_varlist(i)==0 then         eql_initvals(i):=eql_results(1,k);         k:=k+1;       end;     end;   end; end;

The issue was that the source code used the local variable 't' which also happens to be a variable that we declare as a CAS variable via indirection. Fortunately (or unfortunately depending how you view the separation of Home and CAS), the CAS variables seem to be completely separate. Therefore, we can represent the value of the variables as entries within a list we call eql_initvals. Then, by declaring variables (or purging) within the CAS and setting them equal to eql_initvals(i), we do not run into any collision between local and non-local variables.

This scheme can be broken if for some reason we decide to create equations whose variables are also of the same name as those used within our source code! It is my hope, however, that the variable names I used are unique enough that they never appear in any useful equation that people use in real work :-)

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01-10-2015, 10:49 PM
Post: #10
 akmon Member Posts: 197 Joined: Jun 2014
RE: Creating an equation library (updated)
Now it Works fine. Thank you for the new source code.

Another suggestion.
Is it posible to add an equation directly on the program?

I think I´ve left some clues to understand I am a big fan of SOLVESYS library, my mind is forcing to suggest your program to be as similar as posible to that one. My apologies.
The purpose is to have a program that can read a note file with all equations (eqlib), or simply make a fast calculus inserting directly one or several equations for an specific problem (solvesys). All in one code. fsolve is better than solve App for numeric solution (a high percentaje of complex equations problems you just need the final numeric value).
01-10-2015, 11:07 PM
Post: #11
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
(01-10-2015 10:49 PM)akmon Wrote:  Now it Works fine. Thank you for the new source code.

Another suggestion.
Is it posible to add an equation directly on the program?

I think I´ve left some clues to understand I am a big fan of SOLVESYS library, my mind is forcing to suggest your program to be as similar as posible to that one. My apologies.
The purpose is to have a program that can read a note file with all equations (eqlib), or simply make a fast calculus inserting directly one or several equations for an specific problem (solvesys). All in one code. fsolve is better than solve App for numeric solution (a high percentaje of complex equations problems you just need the final numeric value).

Yeah, the idea was to basically make a hybrid of SOLVESYS and the EQLIB from the HP48 days.

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01-13-2015, 03:11 PM
Post: #12
 acmeanvil Junior Member Posts: 19 Joined: Apr 2014
RE: Creating an equation library (updated)
Hi Han,
That is an excellent prototype. I will follow progress closely.

And you are correct, it would be great to be able to incorporate images in with the choose box...
01-13-2015, 04:21 PM
Post: #13
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
The solver portion is done and behaves much like the SOLVESYS solver. The slight difference is that the solver falls back to fsolve in the case where there are more variables than there are equations (usually infinite solutions). In this case, the non-constant initial values are ignored. If there are more equations than variables, then a least squares approach will be used.

What I am working on now is the interface. Sadly, the CHOOSE() and MSGBOX() commands are just not sufficient for what I need. Moreover, they keep switching the view back to the command line view rather than just drawing directly onto the current view.

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01-13-2015, 06:51 PM
Post: #14
 akmon Member Posts: 197 Joined: Jun 2014
RE: Creating an equation library (updated)
Fantastic, I´m waiting for the code to test some linear and no linear system of equations.
01-13-2015, 11:12 PM (This post was last modified: 01-13-2015 11:15 PM by Helge Gabert.)
Post: #15
 Helge Gabert Senior Member Posts: 467 Joined: Dec 2013
RE: Creating an equation library (updated)
Nice project, Han!

Do you plan on having check boxes for variables that are allowed to be complex as well? (The user might want the solution(s) to be restricted to reals in some cases, but at other times not).
01-14-2015, 06:32 AM
Post: #16
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
After some testing, I have run into a snag. Apparently the recent firmware changed how the command SVD outputs its results -- and the results themselves!

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01-14-2015, 05:08 PM
Post: #17
 Helge Gabert Senior Member Posts: 467 Joined: Dec 2013
RE: Creating an equation library (updated)
I noticed that for arbitrary n x m matrices, the supposedly rectangular vector (the second element returned in the SVD output list), has dimension n if n<m, or m if n>m, so you can't just call diag(outputlist(2)) in order to reconstruct.

Is this the problem?

The results, in general, seem to be OK, though.
01-14-2015, 05:45 PM (This post was last modified: 01-14-2015 05:46 PM by Han.)
Post: #18
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
No, that isn't the issue. Try this matrix:

SVD([[3,4.5,4.5],[4.5,12.25,1.25],[4.5,1.25,12.25]]);

Just copy that line and paste it into the command line of the emulator. I get some "noise" (complex-valued matrices and vectors); see attached screenshot. The bug is also present in hardware (latest firmware)

Attached File(s) Thumbnail(s)

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01-14-2015, 09:03 PM (This post was last modified: 01-14-2015 09:33 PM by Helge Gabert.)
Post: #19
 Helge Gabert Senior Member Posts: 467 Joined: Dec 2013
RE: Creating an equation library (updated)
Yes I see what you are saying - the matrix is symmetric, and Wolfram alpha gives a purely real solution. Even with complex checked off in Home settings, and use i checked off in CAS settings it returns the complex elements in the U matrix.

But . . . presume the output is stored in L1, then, in CAS, L1(1)*diag(L1(2))*TRN(L1(3)) returns the original matrix - - so couldn't one argue that the complex SVD output is also correct?
01-14-2015, 11:03 PM
Post: #20
 Han Senior Member Posts: 1,883 Joined: Dec 2013
RE: Creating an equation library (updated)
(01-14-2015 09:03 PM)Helge Gabert Wrote:  Yes I see what you are saying - the matrix is symmetric, and Wolfram alpha gives a purely real solution. Even with complex checked off in Home settings, and use i checked off in CAS settings it returns the complex elements in the U matrix.

But . . . presume the output is stored in L1, then, in CAS, L1(1)*diag(L1(2))*TRN(L1(3)) returns the original matrix - - so couldn't one argue that the complex SVD output is also correct?

Interesting... I didn't even bother to check that particular calculation. What I don't understand is why there is even a complex component when the characteristic polynomial has real roots.

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