source: git/Singular/LIB/ntsolve.lib @ fd3fb7

//(GMG, last modified 16.12.00)
///////////////////////////////////////////////////////////////////////////////
version="$Id: ntsolve.lib,v 1.9 2000-12-19 15:05:31 anne Exp $";
category="Symbolic-numerical solving";
info="
LIBRARY:  ntsolve.lib     REAL NEWTON SOLVING OF POLYNOMIAL SYSTEMS
AUTHORS:  Wilfred Pohl, email: pohl@mathematik.uni-kl.de
          Dietmar Hillebrand

PROCEDURES:
 nt_solve(G,..);        find one real root of 0-dimensional ideal G
 triMNewton(G,..);      find one real root for 0-dim triangular system G
";

///////////////////////////////////////////////////////////////////////////////

proc nt_solve (ideal gls, ideal ini, list #)
"USAGE:   nt_solve(gls,ini[,ipar]); gls ideal, ini ideal, ipar list or intvec
         gls: contains the equations, for which a solution will be computed
         ini: ideal of initial values (approximate solutions to start with)
         ipar: control integers (default: ipar = 100,10)
         ipar[1] - max. number of iterations
         ipar[2] - accuracy (we have the l_2-norm ||.||): accept solution 
         sol if ||gls(sol)|| < eps0*(0.1^ipar[2]) where eps0 = ||gls(ini)|| 
         is the initial error
ASSUME:  gls is a zerodimensional ideal with nvars(basering) = size(gls) (>1)
RETURN:  ideal, coordinates of one solution (if found), 0 else
NOTE:    printlevel >0: displays comments (default =0)
EXAMPLE: example nt_solve; shows an example
"
{
    def rn = basering;
    int di = size(gls);
    if (nvars(basering) != di){
      ERROR("// wrong number of equations");}
    if (size(ini) != di){
      ERROR("// wrong number of initial values");}
    int prec = system("getPrecDigits"); // precision

    int i1,i2,i3;
    int itmax, acc;
    intvec ipar;
    if ( size(#)>0 ){
       i1=1;
       if (typeof(#[1])=="intvec") {ipar=#[1];}
       if (typeof(#[1])=="int") {ipar[1]=#[1];}
       if ( size(#)>1 ){
          i1=2;
          if (typeof(#[2])=="int") {ipar[2]=#[2];}
       }
    }

    int prot = printlevel-voice+2;  // prot=printlevel (default:prot=0)
    if (i1 < 1){itmax = 100;}else{itmax = ipar[1];}
    if (i1 < 2){acc = prec/2;}else{acc = ipar[2];}
    if ((acc <= 0)||(acc > prec-1)){acc = prec-1;}

    int dpl = di+1;
    string out; 
    out = "ring rnewton=(real,prec),("+varstr(basering)+"),(c,dp);";
    execute(out);

    ideal gls1=imap(rn,gls);
    module nt,sub;
    sub = transpose(jacob(gls1));
    for (i1=di;i1>0;i1--){
      if(sub[i1]==0){break;}}
    if (i1>0){
      setring rn; kill rnewton;
      ERROR("// one var not in equation");}

    list direction;
    ideal ini1;
    ini1 = imap(rn,ini);
    number dum,y1,y2,y3,genau;
    genau = 0.1;
    dum = genau;
    genau = genau^acc;

    for (i1=di;i1>0;i1--){
      sub[i1]=sub[i1]+gls1[i1]*gen(dpl);}
    nt = sub;
    for (i1=di;i1>0;i1--){
      nt = subst(nt,var(i1),ini1[i1]);}

    // now we have in sub the general structure
    // and in nt the structure with subst. vars

    // compute initial error
    y1 = ml2norm(nt,genau);
    dbprint(prot,"// initial error = "+string(y1)); 
    y2 = genau*y1;

  // begin of iteration
  for(i3=1;i3<=itmax;i3++){
     dbprint(prot,"// iteration: "+string(i3)); 

    // find newton direction
    direction=bareiss(nt,1,-1);

    // find dumping
    dum = linesearch(gls1,ini1,direction[1],y1,dum,genau);
    if (i3%5 == 0)
    {
      if (dum <= 0.000001)
      {
        dum = 1.0;
      }
    }
    dbprint(prot,"// dumping = "+string(dum));

    // new value
    for(i1=di;i1>0;i1--){
      ini1[i1]=ini1[i1]-dum*direction[1][i1];}
    nt = sub;
    for (i1=di;i1>0;i1--){
      nt = subst(nt,var(i1),ini1[i1]);}
    y1 = ml2norm(nt,genau);
    dbprint(prot,"// error = "+string(y1));
    if(y1<y2){break;} // we are ready
  }

    if (y1>y2){
      "// ** WARNING: iteration bound reached with error > error bound!";}
    setring rn;
    ini = imap(rnewton,ini1);
    kill rnewton;
    return(ini);
}
example
{
    "EXAMPLE:";echo=2;
    ring rsq = (real,40),(x,y,z,w),lp;
    ideal gls =  x2+y2+z2-10, y2+z3+w-8, xy+yz+xz+w5 - 1,w3+y;
    ideal ini = 3.1,2.9,1.1,0.5;
    intvec ipar = 200,0;
    ideal sol = nt_solve(gls,ini,ipar);
    sol;
}
///////////////////////////////////////////////////////////////////////////////

static proc sqrt (number wr, number wa, number wg)
{
  number es,we;
  number wb=wa;
  number wf=wb*wb-wr;
  if(wf>0){
    es=wf;}
  else{
    es=-wf;}
  we=wg*es;
  while (es>we)
  {
    wf=wf/(wb+wb);
    wb=wb-wf;
    wf=wb*wb-wr;
    if(wf>0){
      es=wf;}
    else{
      es=-wf;}
  }
  return(wb);
}

static proc il2norm (ideal H, number wg)
{
  number wa,wb;
  int wi,dpl;
  wa = leadcoef(H[1]);
  wa = wa*wa;
  for(wi=size(H);wi>1;wi--)
  {
    wb=leadcoef(H[wi]);
    wa=wa+wb*wb;
  }
  return(sqrt(wa,wa,wg));
}

static proc ml2norm (module H, number wg)
{
  number wa,wb;
  int wi,dpl;
  dpl = size(H)+1;
  wa = leadcoef(H[1][dpl]);
  wa = wa*wa;
  for(wi=size(H);wi>1;wi--)
  {
    wb=leadcoef(H[wi][dpl]);
    wa=wa+wb*wb;
  }
  return(sqrt(wa,wa,wg));
}

static
proc linesearch(ideal nl, ideal aa, ideal bb,
number z1, number tt, number gg)
{
  int ii,d;
  ideal cc,jn;
  number ss,z2,z3,mm;

  mm=0.000001;
  ss=tt;
  d=size(nl);
  cc=aa;
  for(ii=d;ii>0;ii--){cc[ii]=cc[ii]-ss*bb[ii];}
  jn=nl;
  for(ii=d;ii>0;ii--){jn=subst(jn,var(ii),cc[ii]);}
  z2=il2norm(jn,gg);
  z3=-1;
  while(z2>=z1)
  {
    ss=0.5*ss;
    if(ss<mm){return (mm);}
    cc=aa;
    for(ii=d;ii>0;ii--)
    {
      cc[ii]=cc[ii]-ss*bb[ii];
    }
    jn=nl;
    for(ii=d;ii>0;ii--){jn=subst(jn,var(ii),cc[ii]);}
    z3=z2;
    z2=il2norm(jn,gg);
  }
  if(z3<0)
  {
    while(z3<z2)
    {
      ss=ss+ss;
      cc=aa;
      for(ii=d;ii>0;ii--)
      {
        cc[ii]=cc[ii]-ss*bb[ii];
      }
      jn=nl;
      for(ii=d;ii>0;ii--){jn=subst(jn,var(ii),cc[ii]);}
      if(z3>0){z2=z3;}
      z3=il2norm(jn,gg);
    }
  }
  z2=z2-z1;
  z3=z3-z1;
  ss=0.25*ss*(z3-4*z2)/(z3-2*z2);
  if(ss>1.0){return (1.0);}
  if(ss<mm){return (mm);}
  return(ss);
}
///////////////////////////////////////////////////////////////////////////////
//
//   Multivariate Newton for triangular systems
//   algorithms for solving algebraic system of dimension zero
//   written by Dietmar Hillebrand
///////////////////////////////////////////////////////////////////////////////

proc triMNewton (ideal G, ideal a, list #)
"USAGE:  triMNewton(G,a[,ipar]); G ideal, a ideal, ipar list or intvec
ASSUME:  G:   g1,..,gn, a triangular system of n equations in n vars, 
              i.e gi=gi(var(n-i+1),..,var(n))
         a:   ideal of numbers, coordinates of an approximation of a common 
              zero of G to start with (with a[i] to be substituted in var(i))
         ipar: control integer vector (default: ipar = 100,10)
         ipar[1] - max. number of iterations
         ipar[2] - accuracy (we have as norm |.| absolute value ): 
         accept solution  sol if |G(sol)| < |G(a)|*(0.1^ipar[2])
         itb: int, the maximal number of iterations performed
         err: number, an error bound
RETURN:  an ideal, coordinates of a better approximation of a zero of G
EXAMPLE: example triMNewton; shows an example
"
{
    int prot = printlevel;
    int i1,i2,i3;
    intvec ipar;
    if ( size(#)>0 ){
       i1=1;
       if (typeof(#[1])=="intvec") {ipar=#[1];}
       if (typeof(#[1])=="int") {ipar[1]=#[1];}
       if ( size(#)>1 ){
          i1=2;
          if (typeof(#[2])=="int") {ipar[2]=#[2];}
       }
    }
    int itb, err;
    if (i1 < 1) {itb = 100;} else {itb = ipar[1];}
    if (i1 < 2) {err = 10;} else {err = ipar[2];}
  
    if (itb == 0)
    {
       dbprint(prot,"// ** iteration bound reached with error > error bound!");
       return(a);
    }

    int i,j,k;
    ideal p=G;
    matrix J=jacob(G);
    list h;
    poly hh;
    int fertig=1;
    int n=nvars(basering);

    for (i = 1; i <= n; i++)
    {
        for (j = n; j >= n-i+1; j--)
        {
            p[i] = subst(p[i],var(j),a[j]);
            for (k = n; k >= n-i+1; k--)
            {
                J[i,k] = subst(J[i,k],var(j),a[j]);
            }
        }
        if (J[i,n-i+1] == 0)
        {
            ERROR("// ideal not radical");
            return();
        }

        // solve linear equations
        hh = -p[i];
        for (j = n; j >= n-i+2; j--)
        {
            hh = hh - J[i,j]*h[j];
        }
        h[n-i+1] = number(hh/J[i,n-i+1]);
    }

    for (i = 1; i <= n; i++)
    {
        if ( abs(h[i]) > (1/10)^err)
        {
            fertig = 0;
            break;
        }
    }
    if ( not fertig )
    {
        if (prot > 0)
        {
           "// error:"; print(abs(h[i]));
           "// iterations to be performed: "+string(itb);
        }
        for (i = 1; i <= n; i++)
        {
            a[i] = a[i] + h[i];
        }
        ipar = itb-1,err;
        return(triMNewton(G,a,ipar));
    }
    else
    {
        return(a);
    }
}
example
{ "EXAMPLE:"; echo = 2;
   ring r = (real,30),(z,y,x),(lp);
   ideal i = x^2-1,y^2+x4-3,z2-y4+x-1;
   ideal a = 2,3,4;
   intvec e = 20,10;
   ideal l = triMNewton(i,a,e);
   l;
}
///////////////////////////////////////////////////////////////////////////////

static proc abs( number r)
{
   if (r >= 0)
   {
       return(r);
   }
   else
   {
       return(-r);
   }
}
///////////////////////////////////////////////////////////////////////////////