Annotation of OpenXM_contrib/PHC/Ada/Schubert/pieri_homotopies.adb, Revision 1.1
1.1 ! maekawa 1: with Standard_Complex_Numbers; use Standard_Complex_Numbers;
! 2: with Standard_Random_Numbers; use Standard_Random_Numbers;
! 3: with Standard_Natural_Vectors;
! 4: with Standard_Complex_Polynomials; use Standard_Complex_Polynomials;
! 5: with Numeric_Minor_Equations; use Numeric_Minor_Equations;
! 6: with Determinantal_Systems; use Determinantal_Systems;
! 7: with Specialization_of_Planes; use Specialization_of_Planes;
! 8: with Curves_into_Grassmannian; use Curves_into_Grassmannian;
! 9:
! 10: package body Pieri_Homotopies is
! 11:
! 12: -- AUXILIARIES TO THE QUANTUM CASE :
! 13:
! 14: procedure Multiply ( p : in out Poly; var,deg : natural ) is
! 15:
! 16: -- DESCRIPTION :
! 17: -- Multiplies p with x(var)**deg.
! 18:
! 19: procedure Multiply_Term ( t : in out Term; continue : out boolean ) is
! 20: begin
! 21: t.dg(var) := t.dg(var) + deg;
! 22: continue := true;
! 23: end Multiply_Term;
! 24: procedure Multiply_Terms is new Changing_Iterator(Multiply_Term);
! 25:
! 26: begin
! 27: Multiply_Terms(p);
! 28: end Multiply;
! 29:
! 30: procedure Multiply ( xpm : in out Standard_Complex_Poly_Matrices.Matrix;
! 31: col,var,deg : in natural ) is
! 32:
! 33: -- DESCRIPTION :
! 34: -- Multiplies all polynomial in the column col of xpm with x(var)**deg.
! 35:
! 36: begin
! 37: for i in xpm'range(1) loop
! 38: if xpm(i,col) /= Null_Poly
! 39: then Multiply(xpm(i,col),var,deg);
! 40: end if;
! 41: end loop;
! 42: end Multiply;
! 43:
! 44: procedure Add ( p : in out Poly;
! 45: var : in natural; start : in Complex_Number ) is
! 46:
! 47: -- DESCRIPTION :
! 48: -- Performs p := p + (1-s)*c, where s = x(var) and c = start.
! 49:
! 50: n : constant natural := Number_of_Unknowns(p);
! 51: t : Term;
! 52:
! 53: begin
! 54: t.dg := new Standard_Natural_Vectors.Vector'(1..n => 0);
! 55: t.cf := start;
! 56: Add(p,t); -- p = p + c
! 57: t.dg(var) := 1;
! 58: Sub(p,t); -- p = p + c - c*s = p + c*(1-s)
! 59: Clear(t);
! 60: end Add;
! 61:
! 62: procedure Add ( xpm : in out Standard_Complex_Poly_Matrices.Matrix;
! 63: col,var : in natural; start : in Vector ) is
! 64:
! 65: -- DESCRIPTION :
! 66: -- Adds to every nonzero polynomial in the column col of xpm
! 67: -- the term c*(1-s), where s = x(var) and c = start(i).
! 68:
! 69: begin
! 70: for i in xpm'range(1) loop
! 71: if xpm(i,col) /= Null_Poly
! 72: then Add(xpm(i,col),var,start(i));
! 73: end if;
! 74: end loop;
! 75: end Add;
! 76:
! 77: function Degree1 ( nd : Node; n : natural ) return natural is
! 78:
! 79: -- DESCRIPTION :
! 80: -- Returns the maximum of one and the degree of the first column
! 81: -- of the map that fits the pattern as decribed by pivots in the node.
! 82: -- The parameter n must equal m+p.
! 83:
! 84: d : constant natural := (nd.bottom(1) - nd.top(1))/n;
! 85:
! 86: begin
! 87: if d = 0
! 88: then return 1;
! 89: else return d;
! 90: end if;
! 91: end Degree1;
! 92:
! 93: function Moving_Parameter0 ( n,xk,tk : natural;
! 94: start,target : Complex_Number ) return Poly is
! 95:
! 96: -- DESCRIPTION :
! 97: -- Returns the equation (x-start)*(1-t) + (x-target)*t = 0 that describes
! 98: -- the motion of x from start to target as t goes from 0 to 1.
! 99: -- Note that this equation equals x + (start-target)*t - start = 0.
! 100: -- This is the older version without using a random constant.
! 101:
! 102: -- ON ENTRY :
! 103: -- n total number of variables, continuation parameter t included;
! 104: -- xk index of the moving variable x;
! 105: -- tk index of the continuation parameter
! 106: -- start starting value for x;
! 107: -- target target value for x.
! 108:
! 109: res : Poly;
! 110: t : Term;
! 111:
! 112: begin
! 113: t.dg := new Standard_Natural_Vectors.Vector'(1..n => 0);
! 114: t.cf := -start;
! 115: res := Create(t); -- res = -start
! 116: t.cf := Create(1.0);
! 117: t.dg(xk) := 1;
! 118: Add(res,t); -- res = x - start
! 119: t.dg(xk) := 0;
! 120: t.dg(tk) := 1;
! 121: t.cf := start - target; -- res = x + (start - target)*t - start
! 122: Add(res,t);
! 123: Clear(t);
! 124: return res;
! 125: end Moving_Parameter0;
! 126:
! 127: function Moving_Parameter ( n,xk,tk : natural;
! 128: start,target : Complex_Number ) return Poly is
! 129:
! 130: -- DESCRIPTION :
! 131: -- Returns the equation c*(x-start)*(1-t) + (x-target)*t = 0 for
! 132: -- the motion of x from start to target as t goes from 0 to 1.
! 133: -- This version uses a constant c, randomly generated from within.
! 134:
! 135: -- ON ENTRY :
! 136: -- n total number of variables, continuation parameter t included;
! 137: -- xk index of the moving variable x;
! 138: -- tk index of the continuation parameter t;
! 139: -- start starting value for x;
! 140: -- target target value for x.
! 141:
! 142: res : Poly;
! 143: t : Term;
! 144: c : Complex_Number := Random1;
! 145:
! 146: begin
! 147: t.dg := new Standard_Natural_Vectors.Vector'(1..n => 0);
! 148: t.cf := -c*start;
! 149: res := Create(t); -- res = -c*start
! 150: t.cf := -t.cf;
! 151: t.dg(tk) := 1;
! 152: Add(res,t); -- res = -c*start + c*start*t = -c*start*(1-t)
! 153: t.cf := c;
! 154: t.dg(tk) := 0;
! 155: t.dg(xk) := 1;
! 156: Add(res,t); -- res = -c*start*(1-t) + c*x
! 157: t.cf := -t.cf;
! 158: t.dg(tk) := 1;
! 159: Add(res,t); -- res = -c*start*(1-t) + c*x - c*x*t
! 160: t.cf := Create(1.0); -- = -c*start*(1-t) + c*x*(1-t)
! 161: Add(res,t); -- = c*(1-t)*(x-start)
! 162: t.cf := -target; -- res = c*(1-t)*(x-start) + x*t
! 163: t.dg(xk) := 0;
! 164: Add(res,t); -- res = c*(1-t)*(x-start) + x*t - target*t
! 165: Clear(t); -- = c*(1-t)*(x-start) + (x-target)*t
! 166: return res;
! 167: end Moving_Parameter;
! 168:
! 169: function Constant_Parameter ( n,i : natural; s : Complex_Number )
! 170: return Poly is
! 171:
! 172: -- DESCRIPTION :
! 173: -- Returns the polynomial x_i - s; n equals the number of variables.
! 174:
! 175: res : Poly;
! 176: t : Term;
! 177:
! 178: begin
! 179: t.dg := new Standard_Natural_Vectors.Vector'(1..n => 0);
! 180: t.dg(i) := 1;
! 181: t.cf := Create(1.0);
! 182: res := Create(t);
! 183: t.dg(i) := 0;
! 184: t.cf := -s;
! 185: Add(res,t);
! 186: Clear(t);
! 187: return res;
! 188: end Constant_Parameter;
! 189:
! 190: -- TARGET ROUTINES :
! 191:
! 192: function One_Hypersurface_Pieri_Homotopy
! 193: ( n : natural; nd : Node; expbp : Bracket_Polynomial;
! 194: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 195: planes : VecMat ) return Poly_Sys is
! 196:
! 197: res : Poly_Sys(1..nd.level);
! 198: p : constant natural := nd.p;
! 199: m : constant natural := n-p;
! 200: special : Standard_Complex_Matrices.Matrix(1..n,1..m);
! 201: target,start : Poly;
! 202:
! 203: begin
! 204: case nd.tp is
! 205: when top => special := Special_Plane(m,nd.top);
! 206: -- special := Special_Top_Plane(m,nd.top);
! 207: when bottom => special := Special_Plane(m,nd.bottom);
! 208: -- special := Special_Bottom_Plane(m,nd.bottom);
! 209: when others => null; -- mixed case treated separately
! 210: end case;
! 211: for i in 1..nd.level-1 loop
! 212: res(i) := Expanded_Minors(planes(i).all,xpm,expbp);
! 213: Embed(res(i));
! 214: end loop;
! 215: target := Expanded_Minors(planes(nd.level).all,xpm,expbp);
! 216: start := Expanded_Minors(special,xpm,expbp);
! 217: res(nd.level) := Linear_Homotopy(target,start);
! 218: Clear(target); Clear(start);
! 219: return res;
! 220: end One_Hypersurface_Pieri_Homotopy;
! 221:
! 222: function Two_Hypersurface_Pieri_Homotopy
! 223: ( n : natural; nd : Node; expbp : Bracket_Polynomial;
! 224: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 225: planes : VecMat ) return Poly_Sys is
! 226:
! 227: res : Poly_Sys(1..nd.level);
! 228: p : constant natural := nd.p;
! 229: m : constant natural := n-p;
! 230: top_special : constant Standard_Complex_Matrices.Matrix
! 231: := Special_Plane(m,nd.top);
! 232: -- := Special_Top_Plane(m,nd.top);
! 233: bot_special : constant Standard_Complex_Matrices.Matrix
! 234: := Special_Plane(m,nd.bottom);
! 235: -- := Special_Bottom_Plane(m,nd.bottom);
! 236: top_start,bot_start,top_target,bot_target : Poly;
! 237:
! 238: begin
! 239: for i in 1..nd.level-2 loop
! 240: res(i) := Expanded_Minors(planes(i).all,xpm,expbp);
! 241: Embed(res(i));
! 242: end loop;
! 243: top_target := Expanded_Minors(planes(nd.level).all,xpm,expbp);
! 244: top_start := Expanded_Minors(top_special,xpm,expbp);
! 245: res(nd.level) := Linear_Homotopy(top_target,top_start);
! 246: Clear(top_target); Clear(top_start);
! 247: bot_target := Expanded_Minors(planes(nd.level-1).all,xpm,expbp);
! 248: bot_start := Expanded_Minors(bot_special,xpm,expbp);
! 249: res(nd.level-1) := Linear_Homotopy(bot_target,bot_start);
! 250: Clear(bot_target); Clear(bot_start);
! 251: return res;
! 252: end Two_Hypersurface_Pieri_Homotopy;
! 253:
! 254: function One_General_Pieri_Homotopy
! 255: ( n,ind : natural; nd : Node; bs : Bracket_System;
! 256: start,target : Standard_Complex_Matrices.Matrix;
! 257: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 258: planes : VecMat ) return Link_to_Poly_Sys is
! 259:
! 260: res : Link_to_Poly_Sys;
! 261: nva : constant natural := n*nd.p + 1;
! 262: moving_plane : Standard_Complex_Poly_Matrices.Matrix(1..n,target'range(2))
! 263: := Moving_U_Matrix(nva,start,target);
! 264: moving : Poly_Sys(1..bs'last);
! 265:
! 266: begin
! 267: for i in 1..ind-1 loop
! 268: Concat(res,Polynomial_Equations(planes(i).all,xpm));
! 269: end loop;
! 270: if res /= null
! 271: then for i in res'range loop
! 272: Embed(res(i));
! 273: end loop;
! 274: end if;
! 275: moving := Lifted_Expanded_Minors(moving_plane,xpm,bs);
! 276: Concat(res,moving);
! 277: Standard_Complex_Poly_Matrices.Clear(moving_plane);
! 278: return res;
! 279: end One_General_Pieri_Homotopy;
! 280:
! 281: function Two_General_Pieri_Homotopy
! 282: ( n,ind : natural; nd : Node; top_bs,bot_bs : Bracket_System;
! 283: top_start,top_target,bot_start,bot_target
! 284: : Standard_Complex_Matrices.Matrix;
! 285: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 286: planes : VecMat ) return Link_to_Poly_Sys is
! 287:
! 288: -- DESCRIPTION :
! 289: -- Returns the homotopy for general linear subspace intersections,
! 290: -- in case nd.tp = mixed. The parameter ind indicates the plane
! 291: -- planes(ind) towards the constructed homotopy works.
! 292:
! 293: res : Link_to_Poly_Sys;
! 294: nva : constant natural := n*nd.p + 1;
! 295: top_moving : Standard_Complex_Poly_Matrices.Matrix(1..n,top_target'range(2))
! 296: := Moving_U_Matrix(nva,top_start,top_target);
! 297: bot_moving : Standard_Complex_Poly_Matrices.Matrix(1..n,bot_target'range(2))
! 298: := Moving_U_Matrix(nva,bot_start,bot_target);
! 299: top_movsys : Poly_Sys(1..top_bs'last);
! 300: bot_movsys : Poly_Sys(1..bot_bs'last);
! 301:
! 302: begin
! 303: for i in 1..ind-1 loop
! 304: Concat(res,Polynomial_Equations(planes(i).all,xpm));
! 305: end loop;
! 306: if res /= null
! 307: then for i in res'range loop
! 308: Embed(res(i));
! 309: end loop;
! 310: end if;
! 311: top_movsys := Lifted_Expanded_Minors(top_moving,xpm,top_bs);
! 312: bot_movsys := Lifted_Expanded_Minors(bot_moving,xpm,bot_bs);
! 313: Concat(res,top_movsys);
! 314: Concat(res,bot_movsys);
! 315: Standard_Complex_Poly_Matrices.Clear(top_moving);
! 316: Standard_Complex_Poly_Matrices.Clear(bot_moving);
! 317: return res;
! 318: end Two_General_Pieri_Homotopy;
! 319:
! 320: function One_Quantum_Pieri_Homotopy
! 321: ( n : natural; nd : Node; expbp : Bracket_Polynomial;
! 322: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 323: planes : VecMat; s : Vector ) return Poly_Sys is
! 324:
! 325: -- DESCRIPTION :
! 326: -- Returns the Pieri homotopy that corresponds to the node.
! 327: -- This homotopy is set up to work only in the hypersurface case,
! 328: -- when the type of the node is either top or bottom.
! 329:
! 330: res : Poly_Sys(1..nd.level+1);
! 331: p : constant natural := nd.p;
! 332: m : constant natural := n-p;
! 333: nvars : constant natural := nd.level + p + 2;
! 334: -- nvars = level #equations in the x_ij's
! 335: -- + p because not yet fixed the ones
! 336: -- + 2 for s and t, note that t is continuation parameter
! 337: special : Standard_Complex_Matrices.Matrix(1..n,1..m);
! 338: target,start : Poly;
! 339: map : Standard_Complex_Poly_Matrices.Matrix(1..n,1..p);
! 340:
! 341: begin
! 342: case nd.tp is
! 343: when top => special := Special_Plane(m,Modulo(nd.top,m+p));
! 344: Standard_Complex_Poly_Matrices.Copy(xpm,map);
! 345: Swap(map,nvars-1,nvars);
! 346: when bottom => special := Special_Plane(m,Modulo(nd.bottom,m+p));
! 347: map := xpm;
! 348: when others => null; -- mixed case treated separately
! 349: end case;
! 350: for i in 1..nd.level-1 loop
! 351: declare
! 352: eva : Standard_Complex_Poly_Matrices.Matrix(xpm'range(1),xpm'range(2))
! 353: := Eval(xpm,s(i),Create(1.0));
! 354: begin
! 355: res(i) := Expanded_Minors(planes(i).all,eva,expbp);
! 356: Standard_Complex_Poly_Matrices.Clear(eva);
! 357: end;
! 358: end loop;
! 359: target := Expanded_Minors(planes(nd.level).all,map,expbp);
! 360: start := Expanded_Minors(special,map,expbp);
! 361: res(nd.level) := Linear_Interpolation(target,start,nvars);
! 362: if nd.tp = bottom
! 363: then
! 364: res(nd.level+1)
! 365: := Moving_Parameter(nvars,nvars-1,nvars,Create(1.0),s(nd.level));
! 366: else
! 367: res(nd.level+1)
! 368: := Moving_Parameter(nvars,nvars-1,nvars,Create(1.0),
! 369: (Create(1.0)/s(nd.level)));
! 370: Divide_Common_Factor(res(nd.level),nvars);
! 371: end if;
! 372: if nd.tp = top
! 373: then Standard_Complex_Poly_Matrices.Clear(map);
! 374: end if;
! 375: Clear(target); Clear(start);
! 376: return res;
! 377: end One_Quantum_Pieri_Homotopy;
! 378:
! 379: function Two_Quantum_Pieri_Homotopy
! 380: ( n : natural; nd : Node; expbp : Bracket_Polynomial;
! 381: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 382: planes : VecMat; s : Vector ) return Poly_Sys is
! 383:
! 384: res : Poly_Sys(1..nd.level+2);
! 385: p : constant natural := nd.p;
! 386: m : constant natural := n-p;
! 387: nvars : constant natural := nd.level + p + 2;
! 388: top_special : constant Standard_Complex_Matrices.Matrix
! 389: := Special_Plane(m,Modulo(nd.top,m+p));
! 390: bot_special : constant Standard_Complex_Matrices.Matrix
! 391: := Special_Plane(m,Modulo(nd.bottom,m+p));
! 392: top_start,bot_start,top_target,bot_target : Poly;
! 393: map : Standard_Complex_Poly_Matrices.Matrix(xpm'range(1),xpm'range(2))
! 394: := Insert(xpm,nvars);
! 395:
! 396: begin
! 397: -- first do bottom pivots with s1 = nvars-1
! 398: bot_target := Expanded_Minors(planes(nd.level-1).all,map,expbp);
! 399: bot_start := Expanded_Minors(bot_special,map,expbp);
! 400: res(nd.level-1) := Linear_Interpolation(bot_target,bot_start,nvars+1);
! 401: Divide_Common_Factor(res(nd.level-1),nvars+1);
! 402: res(nd.level+1)
! 403: := Moving_Parameter(nvars+1,nvars-1,nvars+1,Create(1.0),s(nd.level-1));
! 404: Clear(bot_target); Clear(bot_start);
! 405: -- swap s1 with s2 to deal with the fixed equations
! 406: Swap(map,nvars-1,nvars);
! 407: for i in 1..nd.level-2 loop
! 408: declare
! 409: eva : Standard_Complex_Poly_Matrices.Matrix(map'range(1),map'range(2))
! 410: := Eval(map,s(i),Create(1.0));
! 411: begin
! 412: res(i) := Expanded_Minors(planes(i).all,eva,expbp);
! 413: Standard_Complex_Poly_Matrices.Clear(eva);
! 414: end;
! 415: end loop;
! 416: -- swap t with s2 for top pivots, s2 = nvars
! 417: Swap(map,nvars,nvars+1);
! 418: top_target := Expanded_Minors(planes(nd.level).all,map,expbp);
! 419: top_start := Expanded_Minors(top_special,map,expbp);
! 420: res(nd.level) := Linear_Interpolation(top_target,top_start,nvars+1);
! 421: res(nd.level+2)
! 422: := Moving_Parameter(nvars+1,nvars,nvars+1,Create(1.0),
! 423: (Create(1.0)/s(nd.level)));
! 424: Divide_Common_Factor(res(nd.level),nvars+1);
! 425: Clear(top_target); Clear(top_start);
! 426: Standard_Complex_Poly_Matrices.Clear(map);
! 427: return res;
! 428: end Two_Quantum_Pieri_Homotopy;
! 429:
! 430: function One_General_Quantum_Pieri_Homotopy
! 431: ( n,ind : natural; nd : Node; s_mode : natural;
! 432: bs : Bracket_System;
! 433: start,target : Standard_Complex_Matrices.Matrix;
! 434: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 435: planes : VecMat; s : Vector ) return Link_to_Poly_Sys is
! 436:
! 437: res : Link_to_Poly_Sys;
! 438: p : constant natural := nd.p;
! 439: m : constant natural := n-p;
! 440: nvars : constant natural := nd.level + p + 2;
! 441: -- nvars = level #equations in the x_ij's
! 442: -- + p because not yet fixed the ones
! 443: -- + 2 for s and t, note that t is continuation parameter
! 444: eva,map : Standard_Complex_Poly_Matrices.Matrix(1..n,1..p);
! 445: moving_plane : Standard_Complex_Poly_Matrices.Matrix(1..n,target'range(2))
! 446: := Moving_U_Matrix(nvars,start,target);
! 447: moving : Poly_Sys(1..bs'last);
! 448: movpar : Poly_Sys(1..1);
! 449: inddiv : natural;
! 450: starcol : Vector(start'range(1));
! 451: deg1 : constant natural := Degree1(nd,n);
! 452:
! 453: begin
! 454: if s_mode = 0
! 455: then for i in starcol'range loop
! 456: starcol(i) := start(i,1);
! 457: end loop;
! 458: end if;
! 459: -- PART I : fixed equations
! 460: case nd.tp is
! 461: when top => Standard_Complex_Poly_Matrices.Copy(xpm,map);
! 462: Swap(map,nvars-1,nvars);
! 463: when bottom => map := xpm;
! 464: when others => null; -- mixed case treated separately
! 465: end case;
! 466: for i in 1..ind-1 loop
! 467: eva := Eval(xpm,s(i),Create(1.0));
! 468: if s_mode = 0
! 469: then --Multiply(eva,1,nvars-1,deg1);
! 470: --Add(eva,1,nvars-1,starcol);
! 471: Multiply(eva,1,nvars-1,deg1);
! 472: Add(eva,1,nvars,starcol);
! 473: end if;
! 474: Concat(res,Polynomial_Equations(planes(i).all,eva));
! 475: Standard_Complex_Poly_Matrices.Clear(eva);
! 476: end loop;
! 477: if res = null
! 478: then inddiv := 0;
! 479: else inddiv := res'last;
! 480: end if;
! 481: -- PART II : moving equation for the plane
! 482: if s_mode = 2
! 483: then moving := Expanded_Minors(moving_plane,map,bs);
! 484: else eva := Eval(xpm,Create(1.0),Create(0.0));
! 485: if s_mode = 0
! 486: then --Multiply(eva,1,nvars-1,deg1);
! 487: --Add(eva,1,nvars-1,starcol);
! 488: Multiply(eva,1,nvars-1,deg1);
! 489: Add(eva,1,nvars,starcol);
! 490: end if;
! 491: moving := Expanded_Minors(moving_plane,eva,bs);
! 492: Standard_Complex_Poly_Matrices.Clear(eva);
! 493: end if;
! 494: Concat(res,moving);
! 495: if nd.tp = top
! 496: then Standard_Complex_Poly_Matrices.Clear(map);
! 497: for i in inddiv+1..res'last loop
! 498: Divide_Common_Factor(res(i),nvars);
! 499: end loop;
! 500: end if;
! 501: -- PART III : moving equation for the interpolation points
! 502: case s_mode is
! 503: when 0 => -- move s from 0 to 1
! 504: case nd.tp is
! 505: when bottom =>
! 506: movpar(1)
! 507: := Moving_Parameter(nvars,nvars-1,nvars,
! 508: Create(0.0),Create(1.0));
! 509: when others => null;
! 510: end case;
! 511: when 1 => -- leave s constant at 1
! 512: movpar(1) := Constant_Parameter(nvars,nvars-1,Create(1.0));
! 513: when 2 => -- move s from 1 to target
! 514: case nd.tp is
! 515: when top =>
! 516: movpar(1) -- move s from 1 to 1/s_ind (swap of s and t)
! 517: := Moving_Parameter(nvars,nvars-1,nvars,Create(1.0),
! 518: (Create(1.0)/s(ind)));
! 519: when bottom =>
! 520: movpar(1) -- move s from 1 to s_ind
! 521: := Moving_Parameter(nvars,nvars-1,nvars,Create(1.0),s(ind));
! 522: when others => null;
! 523: end case;
! 524: when others => null;
! 525: end case;
! 526: Concat(res,movpar);
! 527: return res;
! 528: end One_General_Quantum_Pieri_Homotopy;
! 529:
! 530: function Two_General_Quantum_Pieri_Homotopy
! 531: ( n,ind : natural; nd : Node; top_bs,bot_bs : Bracket_System;
! 532: top_start,top_target,bot_start,bot_target
! 533: : Standard_Complex_Matrices.Matrix;
! 534: xpm : Standard_Complex_Poly_Matrices.Matrix;
! 535: planes : VecMat; s : Vector ) return Link_to_Poly_Sys is
! 536:
! 537: res : Link_to_Poly_Sys;
! 538:
! 539: begin
! 540: return res;
! 541: end Two_General_Quantum_Pieri_Homotopy;
! 542:
! 543: end Pieri_Homotopies;
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