singular-data 1:4.1.0-p3+ds-2build1 / usr / share / singular / LIB / findifs.lib

//////////////////////////////////////////////////////////////////////////////
version="version findifs.lib 4.0.0.0 Jun_2013 "; // $Id: d56248604e0535012373faef7e0ad53393269302 $
category="System and Control Theory";
info="
LIBRARY: findifs.lib  Tools for the finite difference schemes
AUTHORS: Viktor Levandovskyy,     levandov@math.rwth-aachen.de

OVERVIEW:
        We provide the presentation of difference operators in a polynomial,
        semi-factorized and a nodal form. Running @code{findifs_example();}
        will demonstrate, how we generate finite difference schemes of linear PDEs
        from given approximations.

Theory: The method we use have been developed by V. Levandovskyy and Bernd Martin. The
computation of a finite difference scheme of a given single linear partial
differential equation with constant coefficients with a given approximation
rules boils down to the computation of a Groebner basis of a submodule of
a free module with respect to the ordering, eliminating module components.


Support: SpezialForschungsBereich F1301 of the Austrian FWF

PROCEDURES:
findifs_example();  containes a guided explanation of our approach
decoef(P,n);  decompose polynomial P into summands with respect to the number n
difpoly2tex(S,P[,Q]); present the difference scheme in the nodal form


exp2pt(P[,L]); convert a polynomial M into the TeX format, in nodal form
texcoef(n);  converts the number n into TeX
npar(n); search for 'n' among the parameters and returns its number
magnitude(P); compute the square of the magnitude of a complex expression
replace(s,what,with);  replace in s all the substrings with a given string
xchange(w,a,b); exchange two substrings in a given string

SEE ALSO: latex_lib, finitediff_lib
";


LIB "latex.lib";
LIB "poly.lib";

proc tst_findif()
{
  example decoef;
  example difpoly2tex;
  example exp2pt;
  example texcoef;
  example npar;
  example magnitude;
  example replace;
  example xchange;
}

// static procs:
// par2tex(s);     convert special characters to TeX in s
// mon2pt(P[,L]); convert a monomial M into the TeX format, in nodal form


// 1. GLOBAL ASSUME: in the ring we have first Tx, then Tt: [FIXED, not needed anymore]!
// 2. map vars other than Tx,Tt to parameters instead or just ignore them [?]
// 3. clear the things with brackets
// 4. todo: content resp lcmZ, gcdZ

proc xchange(string where, string a, string b)
"USAGE:  xchange(w,a,b);  w,a,b strings
RETURN:  string
PURPOSE:  exchanges substring 'a' with a substring 'b' in the string w
NOTE:
EXAMPLE: example xchange; shows examples
"{
  // replaces a<->b in where
  // assume they are of the same size [? seems to work]
  string s = "H";
  string t;
  t = replace(where,a,s);
  t = replace(t,b,a);
  t = replace(t,s,b);
  return(t);
}
example
{
 " EXAMPLE:"; echo=2;
 ring r = (0,dt,dh,A),Tt,dp;
 poly p = (Tt*dt+dh+1)^2+2*A;
 string s = texpoly("",p);
 s;
 string t = xchange(s,"dh","dt");
 t;
}

static proc par2tex(string s)
"USAGE:  par2tex(s);  s a string
RETURN:  string
PURPOSE: converts special characters to TeX in s
NOTE: the convention is the following:
      'Tx' goes to 'T_x',
      'dx' to '\\tri x' (the same for dt, dy, dz),
      'theta', 'ro', 'A', 'V' are converted to greek letters.
EXAMPLE: example par2tex; shows examples
"{
  // can be done with the help of latex_lib

// s is a tex string with a poly
// replace theta with \theta
// A with \lambda
// dt with \tri t
// dh with \tri h
// Tx with T_x, Ty with T_y
// Tt with T_t
// V with \nu
// ro with \rho
// dx with \tri x
// dy with \tri y
  string t = s;
  t = replace(t,"Tt","T_t");
  t = replace(t,"Tx","T_x");
  t = replace(t,"Ty","T_y");
  t = replace(t,"dt","\\tri t");
  t = replace(t,"dh","\\tri h");
  t = replace(t,"dx","\\tri x");
  t = replace(t,"dy","\\tri y");
  t = replace(t,"theta","\\theta");
  t = replace(t,"A","\\lambda");
  t = replace(t,"V","\\nu");
  t = replace(t,"ro","\\rho");
  return(t);
}
example
{
 " EXAMPLE:"; echo=2;
 ring r = (0,dt,theta,A),Tt,dp;
 poly p = (Tt*dt+theta+1)^2+2*A;
 string s = texfactorize("",p);
 s;
 par2tex(s);
 string T = texfactorize("",p*(-theta*A));
 par2tex(T);
}

proc replace(string s, string what, string with)
"USAGE:  replace(s,what,with);  s,what,with strings
RETURN:  string
PURPOSE: replaces in 's' all the substrings 'what' with substring 'with'
NOTE:
EXAMPLE: example replace; shows examples
"{
  // clear: replace in s, "what" with "with"
  int ss = size(s);
  int cn = find(s,what);
  if ( (cn==0) || (cn>ss))
  {
    return(s);
  }
  int gn = 0; // global counter
  int sw = size(what);
  int swith = size(with);
  string out="";
  string tmp;
  gn  = 0;
  while(cn!=0)
  {
//    "cn:";    cn;
//    "gn";    gn;
    tmp = "";
    if (cn>gn)
    {
      tmp = s[gn..cn-1];
    }
//    "tmp:";tmp;
//    out = out+tmp+" "+with;
    out = out+tmp+with;
//    "out:";out;
    gn  = cn + sw;
    if (gn>ss)
    {
      // ( (gn>ss) || ((sw>1) && (gn >= ss)) )
      // no need to append smth
      return(out);
    }
//     if (gn == ss)
//     {

//     }
    cn  = find(s,what,gn);
  }
  // and now, append the rest of s
  //  out = out + " "+ s[gn..ss];
  out = out + s[gn..ss];
  return(out);
}
example
{
 " EXAMPLE:"; echo=2;
 ring r = (0,dt,theta),Tt,dp;
 poly p = (Tt*dt+theta+1)^2+2;
 string s = texfactorize("",p);
 s;
 s = replace(s,"Tt","T_t"); s;
 s = replace(s,"dt","\\tri t"); s;
 s = replace(s,"theta","\\theta"); s;
}

proc exp2pt(poly P, list #)
"USAGE:  exp2pt(P[,L]);  P poly, L an optional list of strings
RETURN:  string
PURPOSE: convert a polynomial M into the TeX format, in nodal form
ASSUME: coefficients must not be fractional
NOTE: an optional list L contains a string, which will replace the default
value 'u' for the discretized function
EXAMPLE: example exp2pt; shows examples
"{
// given poly in vars [now Tx,Tt are fixed],
// create Tex expression for points of lattice
// coeffs must not be fractional
  string varnm = "u";
  if (size(#) > 0)
  {
    if (typeof(#[1])=="string")
    {
      varnm = string(#[1]);
    }
  }
  //  varnm;
  string rz,mz;
  while (P!=0)
  {
    mz = mon2pt(P,varnm);
    if (mz[1]=="-")
    {
      rz = rz+mz;
    }
    else
    {
      rz = rz + "+" + mz;
    }
    P = P-lead(P);
  }
  rz  = rz[2..size(rz)];
  return(rz);
}
example
{
  " EXAMPLE:"; echo=2;
  ring r = (0,dh,dt),(Tx,Tt),dp;
  poly M = (4*dh*Tx^2+1)*(Tt-1)^2;
  print(exp2pt(M));
  print(exp2pt(M,"F"));
}

static proc mon2pt(poly M, string V)
"USAGE:  mon2pt(M,V);  M poly, V a string
RETURN:  string
PURPOSE: convert a monomial M into the TeX format, nodal form
EXAMPLE: example mon2pt; shows examples
"{
  // searches for Tx, then Tt
  // monomial to the lattice point conversion
  // c*X^a*Y^b --> c*U^{n+a}_{j+b}
  number cM = leadcoef(M);
  intvec e = leadexp(M);
  //  int a = e[2]; // convention: first Tx, then Tt
  //  int b = e[1];
  int i;
  int a , b, c = 0,0,0;
  int ia,ib,ic = 0,0,0;
  int nv = nvars(basering);
  string s;
  for (i=1; i<=nv ; i++)
  {
    s = string(var(i));
    if (s=="Tt") { a = e[i]; ia = i;}
    if (s=="Tx") { b = e[i]; ib = i;}
    if (s=="Ty") { c = e[i]; ic = i;}
  }
  //  if (ia==0) {"Error:Tt not found!"; return("");}
  //  if (ib==0) {"Error:Tx not found!"; return("");}
  //  if (ic==0)   {"Error:Ty not found!"; return("");}
  //  string tc = texobj("",c); // why not texpoly?
  string tc = texcoef(cM);
  string rs;
  if (cM==-1)
  {
    rs = "-";
  }
  if (cM^2 != 1)
  {
// we don't need 1 or -1 as coeffs
//    rs = clTex(tc)+" ";
//    rs = par2tex(rmDol(tc))+" ";
    rs = par2tex(tc)+" ";
  }
// a = 0 or b = 0
  rs = rs + V +"^{n";
  if (a!=0)
  {
    rs = rs +"+"+string(a);
  }
  rs = rs +"}_{j";
  if (b!=0)
  {
    rs = rs +"+"+string(b);
  }
  if (c!=0)
  {
    rs = rs + ",k+";
    rs = rs + string(c);
  }
  rs = rs +"}";
  return(rs);
}
example
{
 "EXAMPLE:"; echo=2;
  ring r = (0,dh,dt),(Tx,Tt),dp;
  poly M = (4*dh^2-dt)*Tx^3*Tt;
  print(mon2pt(M,"u"));
  poly N = ((dh-dt)/(dh+dt))*Tx^2*Tt^2;
  print(mon2pt(N,"f"));
  ring r2 = (0,dh,dt),(Tx,Ty,Tt),dp;
  poly M  = (4*dh^2-dt)*Tx^3*Ty^2*Tt;
  print(mon2pt(M,"u"));
}

proc npar(number n)
"USAGE:  npar(n);  n a number
RETURN:  int
PURPOSE: searches for 'n' among the parameters and returns its number
EXAMPLE: example npar; shows examples
"{
  // searches for n amongst parameters
  // and returns its number
  int i,j=0,0;
  list L = ringlist(basering);
  list M = L[1][2]; // pars
  string sn = string(n);
  sn = sn[2..size(sn)-1];
  for (i=1; i<=size(M);i++)
  {
    if (M[i] == sn)
    {
      j = i;
    }
  }
  if (j==0)
  {
    "Incorrect parameter";
  }
  return(j);
}
example
{
 "EXAMPLE:"; echo=2;
  ring r = (0,dh,dt,theta,A),t,dp;
  npar(dh);
  number T = theta;
  npar(T);
  npar(dh^2);
}

proc decoef(poly P, number n)
"USAGE:  decoef(P,n);  P a poly, n a number
RETURN:  ideal
PURPOSE:  decompose polynomial P into summands with respect
to the  presence of the number n in the coefficients
NOTE:    n is usually a parameter with no power
EXAMPLE: example decoef; shows examples
"{
  // decomposes polynomial into summands
  // w.r.t. the presence of a number n in coeffs
  // returns ideal
  def br = basering;
  int i,j=0,0;
  int pos = npar(n);
  if ((pos==0) || (P==0))
  {
    return(0);
  }
  pos = pos + nvars(basering);
// map all pars except to vars, provided no things are in denominator
  number con = content(P);
  con = numerator(con);
  P = cleardenom(P); //destroys content!
  P = con*P; // restore the numerator part of the content
  list M = ringlist(basering);
  list L = M[1..4];
  list Pars = L[1][2];
  list Vars = L[2] + Pars;
  L[1] = L[1][1]; // characteristic
  L[2] = Vars;
  // for non-comm things: don't need nc but graded algebra
  //  list templ;
  //  L[5] = templ;
  //  L[6] = templ;
  def @R = ring(L);
  setring @R;
  poly P = imap(br,P);
  poly P0 = subst(P,var(pos),0);
  poly P1 = P - P0;
  ideal I = P0,P1;
  setring br;
  ideal I = imap(@R,I);
  kill @R;
  // check: P0+P1==P
  poly Q = I[1]+I[2];
  if (P!=Q)
  {
    "Warning: problem in decoef";
  }
  return(I);
// substract the pure part from orig and check if n is remained there
}
example
{
 " EXAMPLE:"; echo=2;
  ring r = (0,dh,dt),(Tx,Tt),dp;
  poly P = (4*dh^2-dt)*Tx^3*Tt + dt*dh*Tt^2 + dh*Tt;
  decoef(P,dt);
  decoef(P,dh);
}

proc texcoef(number n)
"USAGE:  texcoef(n);  n a number
RETURN:  string
PURPOSE:  converts the number n into TeX format
NOTE: if n is a polynomial, texcoef adds extra brackets and performs some space substitutions
EXAMPLE: example texcoef; shows examples
"{
  // makes tex from n
  // and uses substitutions
  // if n is a polynomial, adds brackets
  number D  = denominator(n);
  int DenIsOne = 0;
  if ( D==number(1) )
  {
    DenIsOne = 1;
  }
  string sd = texpoly("",D);
  sd = rmDol(sd);
  sd = par2tex(sd);
  number N  = numerator(n);
  string sn = texpoly("",N);
  sn = rmDol(sn);
  sn = par2tex(sn);
  string sout="";
  int i;
  int NisPoly = 0;
  if (DenIsOne)
  {
    sout = sn;
    for(i=1; i<=size(sout); i++)
    {
      if ( (sout[i]=="+") || (sout[i]=="-") )
      {
        NisPoly = 1;
      }
    }
    if (NisPoly)
    {
      sout = "("+sout+")";
    }
  }
  else
  {
    sout = "\\frac{"+sn+"}{"+sd+"}";
  }
  return(sout);
}
example
{
 " EXAMPLE:"; echo=2;
  ring r = (0,dh,dt),(Tx,Tt),dp;
  number n1,n2,n3 = dt/(4*dh^2-dt),(dt+dh)^2, 1/dh;
  n1; texcoef(n1);
  n2; texcoef(n2);
  n3; texcoef(n3);
}

static proc rmDol(string s)
{
  // removes $s and _no_ (s on appearance
  int i = size(s);
  if (s[1] == "$") { s = s[2..i]; i--;}
  if (s[1] == "(") { s = s[2..i]; i--;}
  if (s[i] == "$") { s = s[1..i-1]; i--;}
  if (s[i] == ")") { s = s[1..i-1];}
  return(s);
}

proc difpoly2tex(ideal S, list P, list #)
"USAGE:  difpoly2tex(S,P[,Q]);  S an ideal, P and optional Q are lists
RETURN:  string
PURPOSE: present the difference scheme in the nodal form
ASSUME: ideal S is the result of @code{decoef} procedure
NOTE: a list P may be empty or may contain parameters, which will not
appear in denominators
@* an optional list Q represents the part of the scheme, depending
on other function, than the major part
EXAMPLE: example difpoly2tex; shows examples
"
{
  // S = sum s_i = orig diff poly or
  // the result of decoef
  // P = list of pars (numbers) not to be divided with, may be empty
  // # is an optional list of polys, repr. the part dep. on "f", not on "u"
  //  S = simplify(S,2); // destroys the leadcoef
  // rescan S and remove 0s from it
  int i;
  ideal T;
  int ss = ncols(S);
  int j=1;
  for(i=1; i<=ss; i++)
  {
    if (S[i]!=0)
    {
      T[j]=S[i];
      j++;
    }
  }
  S  = T;
  ss = j-1;
  int GotF = 1;
  list F;
  if (size(#)>0)
  {
    F = #;
    if ( (size(F)==1) && (F[1]==0) )
    {
      GotF = 0;
    }
  }
  else
  {
    GotF = 0;
  }
  int sf = size(F);

  ideal SC;
  int  sp = size(P);
  intvec np;
  int GotP = 1;
  if (sp==0)
  {
    GotP = 0;
  }
  if (sp==1)
  {
    if (P[1]==0)
    {
      GotP = 0;
    }
  }
  if (GotP)
  {
    for (i=1; i<=sp; i++)
    {
      np[i] = npar(P[i])+ nvars(basering);
    }
  }
  for (i=1; i<=ss; i++)
  {
    SC[i] = leadcoef(S[i]);
  }
  if (GotF)
  {
    for (i=1; i<=sf; i++)
    {
      SC[ss+i] = leadcoef(F[i]);
    }
  }
  def br = basering;
// map all pars except to vars, provided no things are in denominator
  list M = ringlist(basering);
  list L = M[1..4]; // erase nc part
  list Pars = L[1][2];
  list Vars = L[2] + Pars;
  L[1] = L[1][1]; // characteristic
  L[2] = Vars;

  def @R = ring(L);
  setring @R;
  ideal SC = imap(br,SC);
  if (GotP)
  {
    for (i=1; i<=sp; i++)
    {
      SC = subst(SC,var(np[i]),1);
    }
  }
  poly q=1;
  q = lcm(q,SC);
  setring br;
  poly  q = imap(@R,q);
  number lq = leadcoef(q);
  //  lq;
  number tmp;
  string sout="";
  string vname = "u";
  for (i=1; i<=ss; i++)
  {
    tmp  = leadcoef(S[i]);
    S[i] = S[i]/tmp;
    tmp = tmp/lq;
    sout = sout +"+ "+texcoef(tmp)+"\\cdot ("+exp2pt(S[i])+")";
  }
  if (GotF)
  {
    vname = "p"; //"f";
    for (i=1; i<=sf; i++)
    {
      tmp  = leadcoef(F[i]);
      F[i] = F[i]/tmp;
      tmp = tmp/lq;
      sout = sout +"+ "+texcoef(tmp)+"\\cdot ("+exp2pt(F[i],vname)+")";
    }
  }
  sout = sout[3..size(sout)]; //rm first +
  return(sout);
}
example
{
  "EXAMPLE:"; echo=2;
  ring r = (0,dh,dt,V),(Tx,Tt),dp;
  poly M = (4*dh*Tx+dt)^2*(Tt-1) + V*Tt*Tx;
  ideal I = decoef(M,dt);
  list L; L[1] = V;
  difpoly2tex(I,L);
  poly G = V*dh^2*(Tt-Tx)^2;
  difpoly2tex(I,L,G);
}


proc magnitude(poly P)
"USAGE:  magnitude(P);  P a poly
RETURN:  poly
PURPOSE: compute the square of the magnitude of a complex expression
ASSUME: i is the variable of a basering
EXAMPLE: example magnitude; shows examples
"
{
  // check whether i is present among the vars
  list L = ringlist(basering)[2]; // vars
  int j; int cnt = 0;
  for(j=size(L);j>0;j--)
  {
    if (L[j] == "i")
    {
      cnt = 1; break;
    }
  }
  if (!cnt)
  {
    ERROR("a variable called i is expected in basering");
  }
  // i is present, check that i^2+1=0;
//   if (NF(i^2+1,std(0)) != 0)
//   {
//     "Warning: i^2+1=0 does not hold. Reduce the output manually";
//   }
  poly re = subst(P,i,0);
  poly im = (P - re)/i;
  return(re^2+im^2);
}
example
{
  "EXAMPLE:"; echo=2;
  ring r = (0,d),(g,i,sin,cos),dp;
  poly P = d*i*sin - g*cos +d^2*i;
  NF( magnitude(P), std(i^2+1) );
}


static proc clTex(string s)
// removes beginning and ending $'s
{
  string t;
  if (size(s)>2)
  {
    // why -3?
     t = s[2..(size(s)-3)];
  }
  return(t);
}

static proc simfrac(poly up, poly down)
{
  // simplifies a fraction up/down
  // into the form up/down = RT[1] + RT[2]/down
  list LL = division(up,down);
  list RT;
  RT[1] =  LL[1][1,1]; // integer part
  RT[2] = L[2][1]; // new numerator
  return(RT);
}

proc findifs_example()
"USAGE:  findifs_example();
RETURN:  nothing (demo)
PURPOSE:  demonstration of our approach and this library
EXAMPLE: example findifs_example; shows examples
"
{

  "* Equation: u_tt - A^2 u_xx -B^2 u_yy = 0; A,B are constants";
  "* we employ three central differences";
  "* the vector we act on is (u_xx, u_yy, u_tt, u)^T";
  "* Set up the ring: ";
  "ring r = (0,A,B,dt,dx,dy),(Tx,Ty,Tt),(c,dp);";
  ring r = (0,A,B,dt,dx,dy),(Tx,Ty,Tt),(c,dp);
  "* Set up the matrix with equation and approximations: ";
  "matrix M[4][4] =";
    "    // direct equation:";
  "    -A^2, -B^2, 1, 0,";
  "    // central difference u_tt";
  "    0, 0,  -dt^2*Tt, (Tt-1)^2,";
  "    // central difference u_xx";
  "    -dx^2*Tx, 0, 0, (Tx-1)^2,";
  "    // central difference u_yy";
  "    0, -dy^2*Ty, 0, (Ty-1)^2;";
  matrix M[4][4] =
    // direct equation:
    -A^2, -B^2, 1, 0,
    // central difference u_tt
    0, 0,  -dt^2*Tt, (Tt-1)^2,
    // central difference u_xx
    -dx^2*Tx, 0, 0, (Tx-1)^2,
    // central difference u_yy
    0, -dy^2*Ty, 0, (Ty-1)^2;
//=========================================
// CHECK THE CORRECTNESS OF EQUATIONS AS INPUT:
ring rX = (0,A,B,dt,dx,dy,Tx,Ty,Tt),(Uxx, Uyy,Utt, U),(c,Dp);
matrix M = imap(r,M);
vector X = [Uxx, Uyy, Utt, U];
 "* Print the differential form of equations: ";
print(M*X);
// END CHECK
//=========================================
 setring r;
 "* Perform the elimination of module components:";
 " module R = transpose(M);";
 " module S = std(R);";
 module R = transpose(M);
 module S = std(R);
 " * See the result of Groebner bases: generators are columns";
 " print(S);";
 print(S);
 " * So, only the first column has its nonzero element in the last component";
 " * Hence, this polynomial is the scheme";
 " poly   p = S[4,1];" ;
 poly   p = S[4,1]; // by elimination of module components
 " print(p); ";
 print(p);
 list L; L[1]=A;L[2] = B;
 ideal I = decoef(p,dt); // make splitting w.r.t. the appearance of dt
 "* Create the nodal of the scheme in TeX format: ";
 " ideal I = decoef(p,dt);";
 " difpoly2tex(I,L);";
 difpoly2tex(I,L); // the nodal form of the scheme in TeX
 "* Preparations for the semi-factorized form: ";
 poly pi1 = subst(I[2],B,0);
 poly pi2 = I[2] - pi1;
 " poly pi1 = subst(I[2],B,0);";
 " poly pi2 = I[2] - pi1;";
 "* Show the semi-factorized form of the scheme: 1st summand";
 " factorize(I[1]); ";
 factorize(I[1]); // semi-factorized form of the scheme: 1st summand
 "* Show the semi-factorized form of the scheme: 2nd summand";
 " factorize(pi1);";
 factorize(pi1); // semi-factorized form of the scheme: 2nd summand
 "* Show the semi-factorized form of the scheme: 3rd summand";
 " factorize(pi1);";
 factorize(pi2); // semi-factorized form of the scheme: 3rd summand
}
example
{
  "EXAMPLE:"; echo=1;
 findifs_example();
}