public MATM3(MATM3 m1) { n = m1.dim(); va = new VECM3[n]; for (int i = 0; i < n; i++) { va[i] = new VECM3(m1[i]); } }
public static MATM3 operator /(MATM3 m1, double a) { int n = m1.dim(); MATM3 m = new MATM3(n); for (int i = 0; i < n; i++) { m[i] = m1[i] / a; } return(m); }
public static MATM3 operator *(double a, MATM3 m1) { int n = m1.dim(); MATM3 m = new MATM3(n); for (int i = 0; i < n; i++) { m[i] = m1[i] * a; } return(m); }
public static MATM3 operator -(MATM3 m1, MATM3 m2) { int n = m1.dim(); MATM3 m = new MATM3(n); for (int i = 0; i < n; i++) { m[i] = m1[i] - m2[i]; } return(m); }
public static MATM3 operator -(MATM3 m1) { int n = m1.dim(); MATM3 A = new MATM3(n); for (int i = 0; i < n; i++) { A[i] = -m1[i]; } return(A); }
public MATM3 tpose() { int dim = this.dim(); MATM3 A = new MATM3(dim); for (int i = 0; i < dim; i++) { for (int j = 0; j < dim; j++) { A[i][j] = new MAT(this[j][i]); } } return(A); }
public static VECV3 fwdSubs(MATM3 m1, VECV3 b) { int len = b.len(); VECV3 v = new VECV3(len); for (int i = 0; i < len; i++) { VEC s = new VEC(3); for (int j = 0; j < i; j++) { s += m1[i][j] * v[j]; } v[i] = (b[i] - s) / m1[i][i]; } return(v); }
public static VECV3 bckSubs(MATM3 m1, VECV3 b) { int len = b.len(); VECV3 v = new VECV3(len); for (int i = 0; i < len; i++) { v[i] = b[i]; } for (int i = len - 1; i >= 0; i--) { v[i] = v[i] / m1[i][i]; for (int j = i - 1; j >= 0; j--) { v[j] = v[j] - (m1[j][i] * v[i]); } } return(v); }
public static MATM3 operator *(MATM3 m1, MATM3 m2) { int n = m1.dim(); MATM3 m = new MATM3(n); MAT s; for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { s = new MAT(3); for (int k = 0; k < n; k++) { s += m1[i][k] * m2[k][j]; } m[i][j] = s; } } return(m); }
public static MATM3 luFact(MATM3 m1) { int n = m1.dim(); for (int i = 0; i < n; i++) { for (int j = i + 1; j < n; j++) { m1[j][i] /= m1[i][i]; } for (int j = i + 1; j < n; j++) { for (int k = i + 1; k < n; k++) { m1[j][k] -= m1[j][i] * m1[i][k]; } } } return(m1); }
public static VECV3 LU_Solve(MATM3 A1, VECV3 b1) { MATM3 A = new MATM3(A1); VECV3 b = new VECV3(b1); int n = A.dim(); A = luFact(A); MATM3 L = new MATM3(n); MATM3 U = new MATM3(n); for (int i = 1; i < n; i++) { for (int j = 0; j < i; j++) { L[i][j] = A[i][j]; } } for (int i = 0; i < n; i++) { L[i][i] = MAT.unit(3); } for (int i = 0; i < n; i++) { for (int j = i; j < n; j++) { U[i][j] = A[i][j]; } } VECV3 y = fwdSubs(L, b); VECV3 x = bckSubs(U, y); return(x); }
/* * void UpdateBackwardMethod(double h) * * Backward method on velocity * Trapezoidal method on position */ public void UpdateBackwardMethod(double h) { if (validCount <= 0) { return; } int[] validIndex = new int[validCount]; int indexTmp = 0; for (int i = 0; i < maxStarNum; i++) { if (validMap[i] == true) { validIndex[indexTmp] = i; indexTmp++; } } VEC[] velAll = new VEC[validCount]; VEC[] new_velAll = new VEC[validCount]; VEC[] posAll = new VEC[validCount]; VEC[] new_posAll = new VEC[validCount]; VEC massAll = new VEC(validCount); for (int i = 0; i < validCount; i++) { velAll[i] = new VEC(s[validIndex[i]].vel); new_velAll[i] = new VEC(s[validIndex[i]].vel); posAll[i] = new VEC(s[validIndex[i]].pos); new_posAll[i] = new VEC(s[validIndex[i]].pos); massAll[i] = s[validIndex[i]].mass; } double e_threshold = 1E-7; double err = 1.0 + e_threshold; int maxIterNum = 100; while (err > e_threshold & maxIterNum > 0) { VECV3 F = new VECV3(2 * validCount); for (int i = 0; i < validCount; i++) { VEC a = new VEC(3); for (int j = 0; j < validCount; j++) { if (i != j) { a += VEC.Normalize(new_posAll[j] - new_posAll[i]) * massAll[j] / DistancePow2(new_posAll[i], new_posAll[j]); } } F[i] = G * a - (new_velAll[i] - velAll[i]) / h; } for (int i = validCount; i < 2 * validCount; i++) { F[i] = (new_posAll[i - validCount] - posAll[i - validCount]) / h - 0.5 * (new_velAll[i - validCount] + velAll[i - validCount]); } MATM3 JF = new MATM3(2 * validCount); MAT tmp; for (int i = 0; i < validCount; i++) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = -1.0 / h; } JF[i][i] = tmp; } for (int i = 0; i < validCount; i++) { for (int j_inJF = validCount; j_inJF < 2 * validCount; j_inJF++) { int j = j_inJF - validCount; tmp = new MAT(3); if (i != j) { for (int a = 0; a < 3; a++) { for (int b = 0; b < 3; b++) { if (a == b) { tmp[a][b] = massAll[j] * G * (-3 * Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -2.5) * (new_posAll[j][a] - new_posAll[i][a]) * new_posAll[j][a] + Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -1.5)); } else { tmp[a][b] = massAll[j] * G * -3 * Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -2.5) * (new_posAll[j][a] - new_posAll[i][a]) * (new_posAll[j][b] - new_posAll[i][b]); } } } } else { for (int a = 0; a < 3; a++) { for (int b = 0; b < 3; b++) { if (a == b) { double sInTmp = 0; for (int k = 0; k < validCount; k++) { if (k != j) { sInTmp += massAll[k] * (3 * Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -2.5) * (new_posAll[k][a] - new_posAll[j][a]) * (new_posAll[k][a] - new_posAll[i][a]) - Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -1.5)); } } tmp[a][a] = G * sInTmp; } else { double sInTmp = 0; for (int k = 0; k < validCount; k++) { if (k != j) { sInTmp += massAll[k] * (3 * Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -2.5) * (new_posAll[k][a] - new_posAll[j][a]) * (new_posAll[k][b] * new_posAll[i][b])); } } tmp[a][a] = G * sInTmp; } } } } JF[i][j_inJF] = tmp; } } for (int i = validCount; i < 2 * validCount; i++) { for (int j = 0; j < validCount; j++) { if (i - validCount == j) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = -0.5; } JF[i][j] = tmp; } } } for (int i = validCount; i < 2 * validCount; i++) { for (int j = validCount; j < 2 * validCount; j++) { if (i == j) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = 1.0 / h; } JF[i][j] = tmp; } } } VECV3 delta = numericalFunctions.LU_Solve(JF, -F); for (int i = 0; i < validCount; i++) { new_velAll[i] += delta[i]; new_posAll[i] += delta[i + validCount]; } err = VECV3.L2_norm_special(F); maxIterNum--; if (maxIterNum == 0) { Debug.Log("Max iter number reached. Err: " + err + " Using Forward method instead in this period."); UpdateForwardMethod(h / 2.0); UpdateForwardMethod(h / 2.0); return; } } for (int i = 0; i < validCount; i++) { if (maxIterNum != 0) { s[validIndex[i]].vel = new_velAll[i]; s[validIndex[i]].pos = new_posAll[i]; } } for (int i = 0; i < validCount; i++) { for (int j = i + 1; j < validCount; j++) { if (DistanceBetweenStarsPow2(s[validIndex[i]], s[validIndex[j]]) < minDistanceToDistroy) { s[validIndex[j]].pos += (s[validIndex[i]].pos - s[validIndex[j]].pos) * s[validIndex[i]].mass / (s[validIndex[j]].mass + s[validIndex[i]].mass); s[validIndex[j]].vel = (s[validIndex[j]].vel * s[validIndex[j]].mass + s[validIndex[i]].vel * s[validIndex[i]].mass) / (s[validIndex[j]].mass + s[validIndex[i]].mass); s[validIndex[j]].mass += s[validIndex[i]].mass; DestroyStar(s[validIndex[i]], 0); continue; } } } }
/* * void UpdateTrapezoidalMethod(double h) * * Trapezoidal method on velocity * Trapezoidal method on position * * Is the most accurate method */ public void UpdateTrapezoidalMethod(double h) { if (validCount <= 0) { return; } // Map valid star index into validIndex int[] validIndex = new int[validCount]; int indexTmp = 0; for (int i = 0; i < maxStarNum; i++) { if (validMap[i] == true) { validIndex[indexTmp] = i; indexTmp++; } } // copy velocity and position of all stars into arrays VEC[] velAll = new VEC[validCount]; VEC[] new_velAll = new VEC[validCount]; VEC[] posAll = new VEC[validCount]; VEC[] new_posAll = new VEC[validCount]; VEC massAll = new VEC(validCount); for (int i = 0; i < validCount; i++) { velAll[i] = new VEC(s[validIndex[i]].vel); new_velAll[i] = new VEC(s[validIndex[i]].vel); posAll[i] = new VEC(s[validIndex[i]].pos); new_posAll[i] = new VEC(s[validIndex[i]].pos); massAll[i] = s[validIndex[i]].mass; } double e_threshold = 1E-9; // the error threshold of performing Newton’s Method double err = 1.0 + e_threshold; int maxIterNum = 100; // Newton’s Method might not converge, we need a max iteration number to terminate. while (err > e_threshold & maxIterNum > 0) { VECV3 F = new VECV3(2 * validCount); for (int i = 0; i < validCount; i++) { VEC a = new VEC(3); for (int j = 0; j < validCount; j++) { if (i != j) { a += 0.5 * VEC.Normalize(new_posAll[j] - new_posAll[i]) * massAll[j] / DistancePow2(new_posAll[i], new_posAll[j]) + 0.5 * VEC.Normalize(posAll[j] - posAll[i]) * massAll[j] / DistancePow2(posAll[i], posAll[j]); } } F[i] = G * a - (new_velAll[i] - velAll[i]) / h; } for (int i = validCount; i < 2 * validCount; i++) { F[i] = (new_posAll[i - validCount] - posAll[i - validCount]) / h - 0.5 * (new_velAll[i - validCount] + velAll[i - validCount]); } MATM3 JF = new MATM3(2 * validCount); MAT tmp; for (int i = 0; i < validCount; i++) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = -1.0 / h; } JF[i][i] = tmp; } for (int i = 0; i < validCount; i++) { for (int j_inJF = validCount; j_inJF < 2 * validCount; j_inJF++) { int j = j_inJF - validCount; tmp = new MAT(3); if (i != j) { for (int a = 0; a < 3; a++) { for (int b = 0; b < 3; b++) { if (a == b) { tmp[a][b] = massAll[j] * G * (-3 * Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -2.5) * (new_posAll[j][a] - new_posAll[i][a]) * new_posAll[j][a] + Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -1.5)); } else { tmp[a][b] = massAll[j] * G * -3 * Math.Pow(DistancePow2(new_posAll[j], new_posAll[i]), -2.5) * (new_posAll[j][a] - new_posAll[i][a]) * (new_posAll[j][b] - new_posAll[i][b]); } } } } else { for (int a = 0; a < 3; a++) { for (int b = 0; b < 3; b++) { if (a == b) { double sInTmp = 0; for (int k = 0; k < validCount; k++) { if (k != j) { sInTmp += massAll[k] * (3 * Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -2.5) * (new_posAll[k][a] - new_posAll[j][a]) * (new_posAll[k][a] - new_posAll[i][a]) - Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -1.5)); } } tmp[a][a] = G * sInTmp; } else { double sInTmp = 0; for (int k = 0; k < validCount; k++) { if (k != j) { sInTmp += massAll[k] * (3 * Math.Pow(DistancePow2(new_posAll[k], new_posAll[j]), -2.5) * (new_posAll[k][a] - new_posAll[j][a]) * (new_posAll[k][b] * new_posAll[i][b])); } } tmp[a][a] = G * sInTmp; } } } } JF[i][j_inJF] = 0.5 * tmp; } } for (int i = validCount; i < 2 * validCount; i++) { for (int j = 0; j < validCount; j++) { if (i - validCount == j) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = -0.5; } JF[i][j] = tmp; } } } for (int i = validCount; i < 2 * validCount; i++) { for (int j = validCount; j < 2 * validCount; j++) { if (i == j) { tmp = new MAT(3); for (int k = 0; k < 3; k++) { tmp[k][k] = 1.0 / h; } JF[i][j] = tmp; } } } VECV3 delta = numericalFunctions.LU_Solve(JF, -F); for (int i = 0; i < validCount; i++) { new_velAll[i] += delta[i]; new_posAll[i] += delta[i + validCount]; } err = VECV3.L2_norm_special(F); maxIterNum--; if (maxIterNum == 0) { Debug.Log("Max iter number reached. Err: " + err + " Using Forward method instead in this period."); for (int i = 0; i < numOfForwardMethodInstead; i++) { UpdateForwardMethod(h / numOfForwardMethodInstead); } return; } } // Apply new position and velocity to Star objects for (int i = 0; i < validCount; i++) { if (maxIterNum != 0) { s[validIndex[i]].vel = new_velAll[i]; s[validIndex[i]].pos = new_posAll[i]; } } // Check if any pair of stars have distance smaller than minDistanceToDistroy. // If so, merge two stars. for (int i = 0; i < validCount; i++) { for (int j = i + 1; j < validCount; j++) { if (DistanceBetweenStarsPow2(s[validIndex[i]], s[validIndex[j]]) < minDistanceToDistroy) { s[validIndex[j]].pos += (s[validIndex[i]].pos - s[validIndex[j]].pos) * s[validIndex[i]].mass / (s[validIndex[j]].mass + s[validIndex[i]].mass); s[validIndex[j]].vel = (s[validIndex[j]].vel * s[validIndex[j]].mass + s[validIndex[i]].vel * s[validIndex[i]].mass) / (s[validIndex[j]].mass + s[validIndex[i]].mass); s[validIndex[j]].mass += s[validIndex[i]].mass; DestroyStar(s[validIndex[i]], 0); continue; } } } }