Annotation of OpenXM_contrib/PHC/Ada/Math_Lib/Supports/floating_linear_inequality_solvers.adb, Revision 1.1
1.1 ! maekawa 1: package body Floating_Linear_Inequality_Solvers is
! 2:
! 3: -- AUXILIARIES :
! 4:
! 5: function Is_In ( v : Standard_Integer_Vectors.Vector;
! 6: i : integer ) return boolean is
! 7:
! 8: -- DESCRIPTION :
! 9: -- Returns true if the number i appears in one of the elements of v.
! 10:
! 11: begin
! 12: for j in v'range loop
! 13: if v(j) = i
! 14: then return true;
! 15: end if;
! 16: end loop;
! 17: return false;
! 18: end Is_In;
! 19:
! 20: function Is_All_In ( v : Standard_Integer_Vectors.Vector; i : integer )
! 21: return boolean is
! 22:
! 23: -- DESCRIPTION :
! 24: -- Returns true if v(k) = i, for k in v'range, false otherwise.
! 25:
! 26: begin
! 27: for k in v'range loop
! 28: if v(k) /= i
! 29: then return false;
! 30: end if;
! 31: end loop;
! 32: return true;
! 33: end Is_All_In;
! 34:
! 35: procedure Copy ( m1 : in Matrix; m2 : out Matrix ) is
! 36:
! 37: -- REQUIRED : ranges must be equal.
! 38:
! 39: -- DESCRIPTION :
! 40: -- Copies the first matrix m1 onto the second matrix m2.
! 41: -- The statement `m2 := m1' does not produce the same result when
! 42: -- compiled with the VADS compiler on IBM RS/6000.
! 43:
! 44: begin
! 45: for i in m1'range(1) loop
! 46: for j in m1'range(2) loop
! 47: m2(i,j) := m1(i,j);
! 48: end loop;
! 49: end loop;
! 50: end Copy;
! 51:
! 52: function Inner_Product ( m : Matrix; i,j : integer ) return double_float is
! 53:
! 54: -- DESCRIPTION :
! 55: -- Returns the inner product of the normals to the ith and jth hyperplane.
! 56:
! 57: res : double_float := 0.0;
! 58:
! 59: begin
! 60: for k in m'first(1)..m'last(1)-1 loop
! 61: res := res + m(k,i)*m(k,j);
! 62: end loop;
! 63: return res;
! 64: end Inner_Product;
! 65:
! 66: -- AUXILIARIES FOR RECURSION ON THE DIMENSION :
! 67:
! 68: function Pivot ( m : Matrix; i : integer ) return integer is
! 69:
! 70: -- DESCRIPTION :
! 71: -- Returns the index of the coefficient with largest absolute value
! 72: -- of the ith inequality.
! 73:
! 74: res : integer := m'first(1);
! 75: max : double_float := abs(m(res,i));
! 76: tmpabs : double_float;
! 77:
! 78: begin
! 79: for j in m'first(1)+1..m'last(1)-1 loop
! 80: tmpabs := abs(m(j,i));
! 81: if tmpabs > max
! 82: then max := tmpabs; res := j;
! 83: end if;
! 84: end loop;
! 85: return res;
! 86: end Pivot;
! 87:
! 88: procedure Eliminate ( m1 : in Matrix; i,j,piv : in integer;
! 89: tol : in double_float; m2 : out Matrix ) is
! 90:
! 91: -- DESCRIPTION :
! 92: -- Performs the elimination on the jth column of the original matrix m1
! 93: -- of inequalities to the reduced matrix m2.
! 94:
! 95: -- REQUIRED : m(piv,i) /= 0 and dimensions of m2 must match.
! 96:
! 97: fac : double_float;
! 98:
! 99: begin
! 100: if abs(m1(piv,j)) < tol
! 101: then for k in m1'range(1) loop
! 102: if k < piv
! 103: then m2(k,j) := m1(k,j);
! 104: elsif k > piv
! 105: then m2(k-1,j) := m1(k,j);
! 106: end if;
! 107: end loop;
! 108: else fac := m1(piv,j)/m1(piv,i);
! 109: for k in m1'range(1) loop
! 110: if k < piv
! 111: then m2(k,j) := m1(k,j) - fac*m1(k,i);
! 112: elsif k > piv
! 113: then m2(k-1,j) := m1(k,j) - fac*m1(k,i);
! 114: end if;
! 115: end loop;
! 116: end if;
! 117: end Eliminate;
! 118:
! 119: function Eliminate ( m : Matrix; mlast2,i,piv : integer;
! 120: tol : in double_float ) return Matrix is
! 121:
! 122: -- DESCRIPTION :
! 123: -- Eliminates one unknown from the system of inequalities.
! 124:
! 125: -- REQUIRED : m(piv,i) /= 0 and m'last(1) > 2.
! 126:
! 127: res : Matrix(m'first(1)..m'last(1)-1,m'first(2)..mlast2);
! 128:
! 129: begin
! 130: for j in res'range(2) loop
! 131: Eliminate(m,i,j,piv,tol,res);
! 132: end loop;
! 133: return res;
! 134: end Eliminate;
! 135:
! 136: function Eliminate ( m : Matrix; mlast2,i : integer; tol : double_float )
! 137: return Matrix is
! 138:
! 139: -- DESCRIPTION :
! 140: -- Computes the pivot for the ith inequality and eliminates one unknown.
! 141: -- The matrix has as columns m'first(2)..mlast2.
! 142:
! 143: -- REQUIRED : Inner_Product(m,i,i) > 0 and m'last(1) > 2.
! 144:
! 145: piv : constant integer := Pivot(m,i);
! 146:
! 147: begin
! 148: return Eliminate(m,mlast2,i,piv,tol);
! 149: end Eliminate;
! 150:
! 151: function Back_Substitute ( m : Matrix; i,piv : integer; x : Vector )
! 152: return Vector is
! 153:
! 154: -- DESCRIPTION :
! 155: -- Returns the back substitution of the vector x which lies on the ith
! 156: -- hyperplane of original inequality system before elimination.
! 157:
! 158: -- REQUIRED : abs(m(piv,i)) > tolerance.
! 159:
! 160: res : Vector(x'first..x'last+1);
! 161:
! 162: begin
! 163: res(res'first..piv-1) := x(x'first..piv-1);
! 164: res(piv+1..res'last) := x(piv..x'last);
! 165: res(piv) := m(m'last(1),i);
! 166: for k in m'first(1)..m'last(1)-1 loop
! 167: if k < piv
! 168: then res(piv) := res(piv) - m(k,i)*x(k);
! 169: elsif k > piv
! 170: then res(piv) := res(piv) - m(k,i)*x(k-1);
! 171: end if;
! 172: end loop;
! 173: res(piv) := res(piv)/m(piv,i);
! 174: return res;
! 175: end Back_Substitute;
! 176:
! 177: -- SELECTORS :
! 178:
! 179: function Evaluate ( m : Matrix; i : integer; x : Vector )
! 180: return double_float is
! 181:
! 182: -- DESCRIPTION :
! 183: -- Evaluates the vector x in the ith inequality.
! 184:
! 185: res : double_float := 0.0;
! 186:
! 187: begin
! 188: for j in x'range loop
! 189: res := res + m(j,i)*x(j);
! 190: end loop;
! 191: return res;
! 192: end Evaluate;
! 193:
! 194: function Satisfies ( m : Matrix; i : integer; x : Vector; tol : double_float )
! 195: return boolean is
! 196: begin
! 197: return (Evaluate(m,i,x) >= m(m'last(1),i) - tol);
! 198: end Satisfies;
! 199:
! 200: function Satisfies ( m : Matrix; x : Vector; tol : double_float )
! 201: return boolean is
! 202: begin
! 203: for i in m'range(2) loop
! 204: if not Satisfies(m,i,x,tol)
! 205: then return false;
! 206: end if;
! 207: end loop;
! 208: return true;
! 209: end Satisfies;
! 210:
! 211: function Satisfies ( m : Matrix; fst,lst : integer;
! 212: x : Vector; tol : double_float ) return boolean is
! 213: begin
! 214: for i in fst..lst loop
! 215: if not Satisfies(m,i,x,tol)
! 216: then return false;
! 217: end if;
! 218: end loop;
! 219: return true;
! 220: end Satisfies;
! 221:
! 222: function First_Violated ( m : Matrix; x : Vector; tol : double_float )
! 223: return integer is
! 224: begin
! 225: for i in m'range(2) loop
! 226: if not Satisfies(m,i,x,tol)
! 227: then return i;
! 228: end if;
! 229: end loop;
! 230: return (m'last(2)+1);
! 231: end First_Violated;
! 232:
! 233: function First_Violated ( m : Matrix; fst,lst : integer;
! 234: x : Vector; tol : double_float ) return integer is
! 235: begin
! 236: for i in fst..lst loop
! 237: if not Satisfies(m,i,x,tol)
! 238: then return i;
! 239: end if;
! 240: end loop;
! 241: return (lst+1);
! 242: end First_Violated;
! 243:
! 244: -- INCONSISTENCY CHECKS :
! 245:
! 246: function Inconsistent ( m : Matrix; i : integer; tol : double_float )
! 247: return boolean is
! 248: begin
! 249: for j in m'first(1)..m'last(1)-1 loop
! 250: if abs(m(j,i)) > tol
! 251: then return false;
! 252: end if;
! 253: end loop;
! 254: return (m(m'last(1),i) > tol);
! 255: end Inconsistent;
! 256:
! 257: function Inconsistent ( m : Matrix; cols : Standard_Integer_Vectors.Vector;
! 258: x : Vector; tol : double_float ) return boolean is
! 259:
! 260: -- ALGORITHM : checks whether the inequality 0 >= c, with c > tol,
! 261: -- can be derived from the combination of the inequalities.
! 262:
! 263: sum : double_float;
! 264:
! 265: begin
! 266: for i in x'range loop -- x should be a positive combination
! 267: if x(i) < 0.0
! 268: then return false;
! 269: end if;
! 270: end loop;
! 271: for i in m'first(1)..m'last(1)-1 loop -- check zero left hand side
! 272: sum := 0.0;
! 273: for j in x'range loop
! 274: sum := sum + x(j)*m(i,cols(j));
! 275: end loop;
! 276: if abs(sum) > tol
! 277: then return false;
! 278: end if;
! 279: end loop;
! 280: sum := 0.0; -- right hand side must be positive
! 281: for j in x'range loop
! 282: sum := sum + x(j)*m(m'last(1),cols(j));
! 283: end loop;
! 284: return (sum > tol);
! 285: end Inconsistent;
! 286:
! 287: function Inconsistent ( m : Matrix; i : integer;
! 288: cols : Standard_Integer_Vectors.Vector; x : Vector;
! 289: tol : double_float ) return boolean is
! 290:
! 291: allcols : Standard_Integer_Vectors.Vector(cols'first..cols'last+1);
! 292: allcoef : Vector(x'first..x'last+1);
! 293:
! 294: begin
! 295: allcols(cols'range) := cols; allcoef(x'range) := x;
! 296: allcols(allcols'last) := i;
! 297: if Is_In(cols,i)
! 298: then allcoef(allcoef'last) := 0.0;
! 299: else allcoef(allcoef'last) := 1.0;
! 300: end if;
! 301: return Inconsistent(m,allcols,allcoef,tol);
! 302: end Inconsistent;
! 303:
! 304: function Inconsistent2D ( m : Matrix; cols : Standard_Integer_Vectors.Vector;
! 305: tol : double_float ) return Vector is
! 306:
! 307: -- REQUIRED : the normals in the corresponding inequality are opposite.
! 308:
! 309: -- DESCRIPTION :
! 310: -- Returns the coefficients in the inconsistency proof.
! 311:
! 312: res : Vector(cols'range) := (cols'range => 1.0);
! 313: f1,f2 : double_float;
! 314:
! 315: begin
! 316: for i in m'first(1)..m'last(1)-1 loop
! 317: f1 := abs(m(i,cols(1)));
! 318: if f1 > tol
! 319: then f2 := abs(m(i,cols(2)));
! 320: if f1 > f2
! 321: then res(2) := 1.0; res(1) := f2/f1;
! 322: else res(1) := 1.0; res(2) := f1/f2;
! 323: end if;
! 324: end if;
! 325: exit when (f1 > tol);
! 326: end loop;
! 327: return res;
! 328: end Inconsistent2D;
! 329:
! 330: function Inconsistent2D ( m : Matrix; i : integer;
! 331: cols : Standard_Integer_Vectors.Vector;
! 332: tol : double_float ) return Vector is
! 333:
! 334: -- REQUIRED : the inequalities of the columns define a nonsingular system.
! 335:
! 336: -- DESCRIPTION :
! 337: -- Returns the coefficients in the inconsistency proof.
! 338:
! 339: res : Vector(cols'range) := (cols'range => 1.0);
! 340: detm12 : constant double_float
! 341: := m(1,cols(1))*m(2,cols(2)) - m(2,cols(1))*m(1,cols(2));
! 342:
! 343: begin
! 344: if abs(detm12) > tol
! 345: then res(1) := (m(2,i)*m(1,cols(2)) - m(1,i)*m(2,cols(2)))/detm12;
! 346: res(2) := (m(1,i)*m(2,cols(1)) - m(2,i)*m(1,cols(1)))/detm12;
! 347: end if;
! 348: return res;
! 349: end Inconsistent2D;
! 350:
! 351: function InconsistentnD ( m : Matrix; i,piv : integer; x : Vector;
! 352: cols : Standard_Integer_Vectors.Vector;
! 353: k : integer; tol : double_float ) return Vector is
! 354:
! 355: -- DESCRIPTION :
! 356: -- Computes the coefficients of the inconsistency proof, based on the
! 357: -- inconsistency proof of the eliminated problem.
! 358:
! 359: res : Vector(x'first..x'last+1);
! 360: fac : double_float;
! 361:
! 362: procedure Compute_Factor is
! 363: begin
! 364: for j in x'range loop
! 365: fac := fac + x(j)*m(piv,cols(j));
! 366: end loop;
! 367: fac := -fac/m(piv,i);
! 368: if abs(fac) > tol
! 369: then for j in x'range loop
! 370: res(j) := x(j)/fac;
! 371: end loop;
! 372: else res(x'range) := x;
! 373: end if;
! 374: end Compute_Factor;
! 375:
! 376: begin
! 377: if Is_In(cols,k)
! 378: then res := (res'range => 0.0);
! 379: if Is_All_In(cols,k)
! 380: then res(res'first) := -m(piv,i)/m(piv,k);
! 381: else fac := 0.0;
! 382: Compute_Factor;
! 383: end if;
! 384: else fac := m(piv,k);
! 385: Compute_Factor;
! 386: if abs(fac) > tol
! 387: then res(res'last) := 1.0/fac;
! 388: else res(res'last) := 1.0;
! 389: end if;
! 390: end if;
! 391: return res;
! 392: end InconsistentnD;
! 393:
! 394: -- CONSTRUCTORS :
! 395:
! 396: procedure Scale ( m : in out Matrix; i : in integer;
! 397: tol : in double_float ) is
! 398:
! 399: ip : double_float := abs(m(Pivot(m,i),i)); --Inner_Product(m,i,i);
! 400:
! 401: begin
! 402: if ip > tol
! 403: then --ip := SQRT(ip);
! 404: for j in m'range(1) loop
! 405: m(j,i) := m(j,i)/ip;
! 406: end loop;
! 407: end if;
! 408: end Scale;
! 409:
! 410: procedure Scale ( m : in out Matrix; tol : in double_float ) is
! 411: begin
! 412: for i in m'range(2) loop
! 413: Scale(m,i,tol);
! 414: end loop;
! 415: end Scale;
! 416:
! 417: procedure Center ( m : in out Matrix; x : in Vector ) is
! 418: begin
! 419: for i in m'range(2) loop
! 420: m(m'last(1),i) := m(m'last(1),i) - Evaluate(m,i,x);
! 421: end loop;
! 422: -- put_line("The centered inequality system : "); put(m,3,3,3);
! 423: end Center;
! 424:
! 425: function Center ( m : Matrix; x : Vector ) return Matrix is
! 426:
! 427: mx : Matrix(m'range(1),m'range(2)); -- := m; problems on RS/6000 ??
! 428:
! 429: begin
! 430: Copy(m,mx);
! 431: Center(mx,x);
! 432: return mx;
! 433: end Center;
! 434:
! 435: procedure Intersect2D ( m : in Matrix; i,j : in integer;
! 436: tol : in double_float;
! 437: x : out Vector; sing : out boolean ) is
! 438:
! 439: detmij : constant double_float := m(1,i)*m(2,j) - m(2,i)*m(1,j);
! 440:
! 441: begin
! 442: if abs(detmij) <= tol
! 443: then sing := true;
! 444: else sing := false;
! 445: x(1) := (m(3,i)*m(2,j) - m(2,i)*m(3,j))/detmij;
! 446: x(2) := (m(1,i)*m(3,j) - m(3,i)*m(1,j))/detmij;
! 447: end if;
! 448: -- put(" i : "); put(i,1); put(" j : "); put(j,1);
! 449: -- put(" detmij : "); put(detmij,3,3,3); new_line;
! 450: end Intersect2D;
! 451:
! 452: -- SOLVE BY INTERSECTION :
! 453:
! 454: procedure Solve_Intersect2D
! 455: ( m : in Matrix; i : in integer; tol : in double_float;
! 456: x : in out Vector; fail : out boolean;
! 457: cols : out Standard_Integer_Vectors.Vector ) is
! 458:
! 459: -- DESCRIPTION :
! 460: -- Enumerates all intersection points of the ith hyperplane with all the
! 461: -- other previous ones. The enumeration stops when either the current
! 462: -- intersection point satisfies the system of inequalities or when the
! 463: -- inequalities for the inconsistency proof are detected.
! 464:
! 465: -- REQUIRED : m'last(1) = 3 and the inequalities are centered.
! 466:
! 467: firstviol,j : integer;
! 468: ffail,sing : boolean := true;
! 469: columns : Standard_Integer_Vectors.Vector(x'range) := (x'range => 0);
! 470:
! 471: begin
! 472: j := m'first(2);
! 473: loop
! 474: Intersect2D(m,j,i,tol,x,sing);
! 475: if not sing
! 476: then firstviol := First_Violated(m,m'first(2),i-1,x,tol);
! 477: if firstviol < j
! 478: then columns(1) := firstviol; columns(2) := j; sing := true;
! 479: x := Inconsistent2D(m,i,columns,tol);
! 480: end if;
! 481: else if Inner_Product(m,i,j) < -tol
! 482: then sing := false; x := (x'range => 0.0);
! 483: for k in m'first(1)..m'last(1)-1 loop
! 484: if abs(m(k,i)) > tol
! 485: then x(k) := m(3,i)/m(k,i);
! 486: if m(k,i) > 0.0
! 487: then sing := ((x(k) - m(3,j)/m(k,j)) > tol);
! 488: else sing := ((m(3,j)/m(k,j) - x(k)) > tol);
! 489: end if;
! 490: end if;
! 491: exit when (abs(m(k,i)) > tol);
! 492: end loop;
! 493: if sing
! 494: then columns(1) := j; columns(2) := i;
! 495: x := Inconsistent2D(m,columns,tol);
! 496: end if;
! 497: else sing := false;
! 498: end if;
! 499: if sing
! 500: then firstviol := j;
! 501: else firstviol := j+1;
! 502: end if;
! 503: end if;
! 504: ffail := (firstviol < i);
! 505: exit when not ffail or sing;
! 506: j := firstviol;
! 507: end loop;
! 508: fail := ffail;
! 509: cols := columns;
! 510: end Solve_Intersect2D;
! 511:
! 512: procedure Solve_IntersectnD
! 513: ( m : in Matrix; i : in integer; tol : in double_float;
! 514: x : in out Vector; fail : out boolean;
! 515: cols : out Standard_Integer_Vectors.Vector ) is
! 516:
! 517: -- DESCRIPTION :
! 518: -- Eliminates one unknown by intersecting with the ith hyperplane.
! 519:
! 520: -- REQUIRED : m'last(1) > 3 and the inequalities are centered.
! 521:
! 522: m2 : Matrix(m'first(1)..m'last(1)-1,m'first(2)..i-1);
! 523: piv : constant integer := Pivot(m,i);
! 524: x2 : Vector(x'first..x'last-1);
! 525: cols2 : Standard_Integer_Vectors.Vector(x2'range);
! 526: k2 : integer;
! 527:
! 528: begin
! 529: if abs(m(piv,i)) > tol
! 530: then m2 := Eliminate(m,i-1,i,piv,tol);
! 531: x2 := (x2'range => 0.0);
! 532: Solve(m2,tol,x2,k2,cols2);
! 533: if k2 >= i
! 534: then fail := false;
! 535: x := Back_Substitute(m,i,piv,x2);
! 536: else fail := true;
! 537: cols(cols2'range) := cols2;
! 538: cols(cols2'last+1) := k2;
! 539: x := InconsistentnD(m,i,piv,x2,cols2,k2,tol);
! 540: end if;
! 541: end if;
! 542: end Solve_IntersectnD;
! 543:
! 544: procedure Solve_Intersect
! 545: ( m : in Matrix; i : in integer; tol : in double_float;
! 546: x : in out Vector; fail : out boolean;
! 547: cols : out Standard_Integer_Vectors.Vector ) is
! 548:
! 549: -- DESCRIPTION :
! 550: -- Intersects the inequalities with the ith hyperplane.
! 551:
! 552: -- REQUIRED : the inequalities are centered,
! 553: -- the ith inequality is first one that is violated.
! 554:
! 555: begin
! 556: if m'last(1) = 3
! 557: then Solve_Intersect2D(m,i,tol,x,fail,cols);
! 558: elsif m'last(1) > 3
! 559: then Solve_IntersectnD(m,i,tol,x,fail,cols);
! 560: end if;
! 561: end Solve_Intersect;
! 562:
! 563: procedure Solve ( m : in Matrix; i : in integer; tol : in double_float;
! 564: x : in out Vector; fail : out boolean;
! 565: cols : out Standard_Integer_Vectors.Vector ) is
! 566:
! 567: inx,xx : Vector(x'range) := (x'range => 0.0);
! 568: wrk : Matrix(m'range(1),m'range(2));
! 569: sing : boolean := true;
! 570: fac : double_float;
! 571:
! 572: begin
! 573: if abs(Inner_Product(m,i,i)) < tol
! 574: then if m(m'last(1),i) > tol
! 575: then fail := true; cols := (x'range => i);
! 576: else fail := false;
! 577: end if;
! 578: else if i = m'first(2)
! 579: then x := (x'range => 0.0);
! 580: for j in x'range loop
! 581: if abs(m(j,i)) > tol
! 582: then sing := false;
! 583: x(j) := m(m'last(1),i)/m(j,i);
! 584: end if;
! 585: exit when not sing;
! 586: end loop;
! 587: fail := sing;
! 588: else Copy(m,wrk);
! 589: if x /= inx
! 590: then Center(wrk,x);
! 591: end if;
! 592: fac := wrk(wrk'last(1),i)/Inner_Product(wrk,i,i);
! 593: for j in inx'range loop
! 594: inx(j) := fac*wrk(j,i);
! 595: end loop;
! 596: if First_Violated(wrk,inx,tol) > i
! 597: then xx := x + inx;
! 598: if Satisfies(m,xx,tol)
! 599: then sing := false;
! 600: end if;
! 601: end if;
! 602: if not sing
! 603: then fail := false; x := xx;
! 604: else Solve_Intersect(wrk,i,tol,inx,sing,cols);
! 605: fail := sing;
! 606: if not sing
! 607: then Add(x,inx);
! 608: else x := inx;
! 609: end if;
! 610: end if;
! 611: end if;
! 612: end if;
! 613: end Solve;
! 614:
! 615: procedure Solve ( m : in Matrix; tol : in double_float; x : in out Vector;
! 616: k : out integer;
! 617: cols : out Standard_Integer_Vectors.Vector ) is
! 618:
! 619: indviol : integer := First_Violated(m,x,tol);
! 620: fail : boolean := false;
! 621:
! 622: begin
! 623: while indviol <= m'last(2) loop
! 624: Solve(m,indviol,tol,x,fail,cols);
! 625: exit when fail;
! 626: indviol := First_Violated(m,indviol+1,m'last(2),x,tol);
! 627: end loop;
! 628: if fail
! 629: then k := indviol;
! 630: else k := m'last(2) + 1;
! 631: end if;
! 632: end Solve;
! 633:
! 634: procedure Iterated_Solve ( m : in Matrix; tol : in double_float;
! 635: x : in out Vector; k : out integer;
! 636: cols : out Standard_Integer_Vectors.Vector ) is
! 637:
! 638: indviol : integer := First_Violated(m,x,tol);
! 639: fail : boolean := false;
! 640: continue : boolean := true;
! 641:
! 642: begin
! 643: while indviol <= m'last(2) loop
! 644: Report(x,indviol-1,continue);
! 645: exit when not continue;
! 646: Solve(m,indviol,tol,x,fail,cols);
! 647: exit when fail;
! 648: indviol := First_Violated(m,indviol+1,m'last(2),x,tol);
! 649: end loop;
! 650: if fail
! 651: then k := indviol;
! 652: else k := m'last(2) + 1;
! 653: end if;
! 654: end Iterated_Solve;
! 655:
! 656: end Floating_Linear_Inequality_Solvers;
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