/// <summary>Computes the undual of <paramref name="A"/> with respect to the whole space. The
/// space has the dimension specified by the metric <paramref name="M"/>.</summary>
/// <param name="M">The metric to be used</param>
/// <returns>dual of <param name="A"/></returns>
public static Multivector Undual(Multivector A, Metric M)
{
Multivector I = Multivector.GetPseudoscalar(M.EigenMetric.Length);
return ip(A, I, M, BasisBlade.InnerProductType.LEFT_CONTRACTION);
}
/// <summary>
/// Computes versor inverse of <paramref name="A"/>.
/// </summary>
/// <param name="A">Multivector to be inverted.</param>
/// <param name="M">The metric to be used for the inverse.</param>
/// <returns>Inverse of versor <paramref name="A"/></returns>
/// <remarks>Computed using the versor inverse method, so <paramref name="A"/>must be a versor or a blade.</remarks>
public static Multivector VersorInverse(Multivector A, Metric M)
{
Multivector R = A.Reverse();
Multivector s = scp(A, R, M);
return VersorInverseInternal(R, s);
}
/// <summary>
/// Computes the scalar product of two multivectors in arbitary non-Euclidean metric.
/// </summary>
/// <param name="A">input Multivector</param>
/// <param name="B">input Multivector</param>
/// <param name="M">metric to be used</param>
public static Multivector ScalarProduct(Multivector A, Multivector B, Metric M)
{
return InnerProduct(A, B, M, BasisBlade.InnerProductType.SCALAR_PRODUCT).ScalarPart();
}
/// <summary>Shortcut to ScalarProduct() </summary>
public static Multivector scp(Multivector A, Multivector B, Metric M)
{
return ScalarProduct(A, B, M);
}
/// <summary>Shortcut to ScalarProduct() </summary>
public static ArrayList scp(BasisBlade A, BasisBlade B, Metric M)
{
return ScalarProduct(A, B, M);
}
/// <summary>Computes the `reverse norm' of this multivector (this.reverse(this)) given a specific metric <paramref name="M"/></paramref>.</summary>
/// <param name="M">The metric to be used in the inner product.</param>
/// <returns>`reverse norm' of this Multivector in specified metric 'M' (can be negative)</returns>
/// <remarks>The reverse norm is s*sqrt(s * this.reverse(this)), where 's' is either +1 or -1,
/// such that the square root is always performed on a number >= 0.</remarks>
public Multivector Norm_r(Metric M)
{
//M.IsPositiveDefinite
return Norm(Multivector.scp(this, Reverse(), M), M.IsPositiveDefinite());
}
/// <summary>
/// Computes the geometric product of two basis blades in arbitary non-Euclidean metric.
/// </summary>
/// <param name="A">Multivector input (LHS)</param>
/// <param name="B">Multivector input (RHS)</param>
/// <param name="M">Metric to be used</param>
/// <returns>geometric product of <paramref name="A"/> and <paramref name="B"/></returns>
public static Multivector GeometricProduct(Multivector A, Multivector B, Metric M)
{
ArrayList L = new ArrayList();
foreach (BasisBlade a in A.BasisBlades)
foreach (BasisBlade b in B.BasisBlades)
{
L.AddRange(BasisBlade.GeometricProduct(a, b, M));
}
return new Multivector(BasisBlade.Simplify(L));
}
/// <summary>Shortcut to InnerProduct() </summary>
public static ArrayList ip(BasisBlade A, BasisBlade B, Metric M, InnerProductType type)
{
return InnerProduct(A, B, M, type);
}
/// <summary>
/// Computes versor inverse of this.
/// </summary>
/// <param name="M">The metric to be used for the inverse</param>
/// <returns>Inverse of this versor</returns>
/// <remarks>Computed using the versor inverse method, so <paramref name="A"/>must be a versor or a blade.</remarks>
public Multivector VersorInverse(Metric M)
{
return VersorInverse(this, M);
}
/// <summary>Computes the dual of <paramref name="A"/> with respect to the whole space. The
/// space has the dimension specified by the metric <paramref name="M"/>.</summary>
/// <param name="M">The metric to be used</param>
/// <returns>dual of <param name="A"/></returns>
public static Multivector Dual(Multivector A, Metric M)
{
Multivector Ii = Multivector.GetPseudoscalar(M.EigenMetric.Length).VersorInverse(M);
return ip(A, Ii, M, BasisBlade.InnerProductType.LEFT_CONTRACTION);
}
/// <summary>Computes unit version of this multivector using Reverse norm.</summary>
/// <returns>normalized Multivector, according to specific metric/norm</returns>
/// <remarks>Throws exception when norm is 0.
/// If the norm is negative, the norm of the returned Multivector will be -1</remarks>
public Multivector Unit_r(Metric M)
{
return Unit(Norm_r(M));
}
/// <summary>Computes the undual of <paramref name="A"/> with respect to the whole space. The
/// space has the dimension specified by the metric <paramref name="M"/>.</summary>
/// <param name="M">The metric to be used</param>
/// <returns>dual of this</returns>
public Multivector Undual(Metric M)
{
return Multivector.Undual(this, M);
}
/// <summary>Computes the squared `reverse norm' of this multivector (this.reverse(this)) given a specific metric <paramref name="m"/></paramref>.</summary>
/// <param name="M">The metric to be used in the inner product.</param>
/// <returns>squared `reverse norm' of this Multivector in specified metric 'm' (can be negative).</returns>
public Multivector Norm_r2(Metric M)
{
return Multivector.scp(this, Reverse(), M);
}
/// <summary>Computes the geometric product of two basis blades in arbitary non-Euclidean metric.</summary>
/// <param name="_A">Basisblade</param>
/// <param name="_B">Basisblade</param>
/// <param name="M">Metric to be used</param>
/// <returns>An ArrayList, because the result may not be a single BasisBlade anymore. For example,
/// <c>no ni = no.ni + no^ni = -1 + no^ni</c>. </returns>
public static ArrayList GeometricProduct(BasisBlade _A, BasisBlade _B, Metric M)
{
// convert arguments to 'eigenbasis'
ArrayList A = M.ToEigenbasis(_A);
ArrayList B = M.ToEigenbasis(_B);
ArrayList result = new ArrayList();
double[] EM = M.EigenMetric;
foreach (BasisBlade bA in A)
foreach (BasisBlade bB in B)
{
BasisBlade C = GeometricProduct(bA, bB, EM);
if (C.scale != 0.0) result.Add(C);
}
return M.ToMetricBasis(result);
}
/// <summary>Shortcut to GeometricProduct() </summary>
public static Multivector gp(Multivector A, Multivector B, Metric M)
{
return GeometricProduct(A, B, M);
}
/// <summary>
/// Computes the inner product of two basis blades in Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="M">The Metric to be used.</param>
/// <param name="type">inner product type to be used. Use one of the InnerProductType constants:
/// LEFT_CONTRACTION, RIGHT_CONTRACTION, HESTENES_INNER_PRODUCT, MODIFIED_HESTENES_INNER_PRODUCT or SCALAR_PRODUCT.</param>
/// <returns>Inner product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static ArrayList InnerProduct(BasisBlade A, BasisBlade B, Metric M, InnerProductType type)
{
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, M), type);
}
/// <summary>
/// Computes the inner product of two multivectors in arbitary non-Euclidean metric.
/// </summary>
/// <param name="A">input Multivector</param>
/// <param name="B">input Multivector</param>
/// <param name="M">metric to be used</param>
/// <param name="type">inner product type to be used</param>
/// <returns>Inner product of <paramref name="A"/> and <paramref name="B"/></returns>
public static Multivector InnerProduct(Multivector A, Multivector B, Metric M, BasisBlade.InnerProductType type)
{
ArrayList L = new ArrayList();
foreach (BasisBlade a in A.BasisBlades)
foreach (BasisBlade b in B.BasisBlades)
{
L.AddRange(BasisBlade.InnerProduct(a, b, M, type));
}
return new Multivector(BasisBlade.Simplify(L));
}
/// <summary>
/// Computes the inner product of two basis blades in arbitary non-Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="M">The Metric to be used</param>
/// <returns>Scalar product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static ArrayList ScalarProduct(BasisBlade A, BasisBlade B, Metric M)
{
InnerProductType type = InnerProductType.SCALAR_PRODUCT;
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, M), type);
}
/// <summary>Shortcut to InnerProduct() </summary>
public static Multivector ip(Multivector A, Multivector B, Metric M, BasisBlade.InnerProductType type)
{
return InnerProduct(A, B, M, type);
}
static void Main(string[] args)
{
System.Console.WriteLine("Hello world");
{
ArrayList AL = new ArrayList();
AL.Add(new BasisBlade(1, 1.0, "a1"));
AL.Add(new BasisBlade(2, 1.0, "a2"));
Multivector A = new Multivector(AL);
System.Console.WriteLine("A = " + A.ToString());
ArrayList BL = new ArrayList();
BL.Add(new BasisBlade(2, 1.0, "b1"));
BL.Add(new BasisBlade(3, 1.0, "b2"));
Multivector B = new Multivector(BL);
System.Console.WriteLine("B = " + B.ToString());
Multivector AB = Multivector.gp(A, B);
System.Console.WriteLine("AB = " + AB.ToString());
Symbolic.HashtableSymbolicEvaluator HSE = new Symbolic.HashtableSymbolicEvaluator();
HSE.Add("a1", 3.0);
HSE.Add("a2", -2.0);
HSE.Add("b1", 4.0);
HSE.Add("b2", 1.0);
Multivector ABeval = AB.SymbolicEval(HSE);
System.Console.WriteLine("ABeval = " + ABeval.ToString());
return;
}
Metric M;
try
{
// Conformal metric (no, e1, e2, e3, ni)
/* int N = 5;
double[] EVM = new double[]{
-0.707107, -0.707107, 0, 0, 0,
0, 0, 0, 0, 1,
0, 0, 1, 0, 0,
0, 0, 0, -1, 0,
-0.707107, 0.707107, 0, 0, 0};
double[] EV = new double[] { -1.0, 1.0, 1.0, 1.0, 1.0 };*/
/*// Euclidean metric (e1, e2, e3)
double[] m = new double[]{
1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
0.0, 0.0, 1.0};*/
// Conformal metric (no, e1, e2, e3, ni)
double[] m = new double[]{
0.0, 0.0, 0.0, 0.0, -1.0,
0.0, 1.0, 0.0, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0, 0.0,
-1.0, 0.0, 0.0, 0.0, 0.0};
M = new Metric(m);
}
catch (Exception E)
{
System.Console.WriteLine(E.ToString());
return;
}
/*{
ArrayList AL = new ArrayList();
AL.Add(new BasisBlade(1 << 3)); // e1
Multivector A = new Multivector(AL);
ArrayList BL = new ArrayList();
BL.Add(new BasisBlade(1 << 3)); // e1
Multivector B = new Multivector(BL);
Multivector V = Multivector.InnerProduct(A, B, M, BasisBlade.InnerProductType.LEFT_CONTRACTION);
System.Console.WriteLine("A = " + A.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("B = " + B.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("A . B = " + V.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
}*/
/* ArrayList AL = new ArrayList();
// AL.Add(new BasisBlade(2 + 16, 1.0, "a_e1ni"));
// AL.Add(new BasisBlade(4 + 16, 1.0, "a_e2ni"));
// AL.Add(new BasisBlade(8 + 16, 1.0, "a_e3ni"));
// AL.Add(new BasisBlade(2 + 4, 1.0, "a_e1e2"));
// AL.Add(new BasisBlade(4 + 8, 1.0, "a_e2e3"));
// AL.Add(new BasisBlade(8 + 2, 1.0, "a_e1e3"));
// AL.Add(new BasisBlade(1 + 16, 1.0, "a_noni"));
AL.Add(new BasisBlade(2 + 16, 1.0));
AL.Add(new BasisBlade(4 + 16, 2.0));
AL.Add(new BasisBlade(8 + 16, 3.0));
AL.Add(new BasisBlade(2 + 4, 4.0));
AL.Add(new BasisBlade(4 + 8, 5.0));
AL.Add(new BasisBlade(8 + 2, 6.0));
Multivector A = new Multivector(AL);
Multivector Aexp = A.Exp(M, false, 0.0);
System.Console.WriteLine("A = " + A.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }) + ",");
System.Console.WriteLine("expA = " + Aexp.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }) + ",");*/
// Symbolic.HashtableSymbolicEvaluator HSE = new Symbolic.HashtableSymbolicEvaluator();
// HSE.Add("a_e1ni", 1.0);
// HSE.Add("a_e2ni", 2.0);
// HSE.Add("a_e3ni", 3.0);
// HSE.Add("a_e1e2", 4.0);
// HSE.Add("a_e2e3", 5.0);
// HSE.Add("a_e1e3", 6.0);
// HSE.Add("a_noni", 6.0);
// System.Console.WriteLine("A = " + A.SymbolicEval(HSE).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }) + ",");
// System.Console.WriteLine("expA = " + Aexp.SymbolicEval(HSE).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }) + ",");
/* // compute square
Multivector A2 = Multivector.gp(A, A, M);
System.Console.WriteLine("A2 = " + A2.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
// evaluate with random coords
Symbolic.RandomSymbolicEvaluator RSE = new Symbolic.RandomSymbolicEvaluator(-100.0, 100.0);
int pos = 0, neg = 0, zero = 0;
for (int i = 0; i < 1; i++)
{
Multivector MV = A2.SymbolicEval(RSE);
double val = MV.RealScalarPart();
System.Console.WriteLine("MV = " + MV);
System.Console.WriteLine("val = " + val);
if (val > 1e-7) pos++;
else if (val < -1e-7) neg++;
else zero++;
RSE.reset();
}
System.Console.WriteLine("pos = " + pos);
System.Console.WriteLine("neg = " + neg);
System.Console.WriteLine("zero = " + zero);*/
/*
ArrayList AL = new ArrayList();
AL.Add(new BasisBlade(1 << 0, 1.0, "a_e1"));
AL.Add(new BasisBlade(1 << 1, 1.0, "a_e2"));
AL.Add(new BasisBlade(1 << 2, 1.0, "a_e3"));
Multivector A = new Multivector(AL);
ArrayList BL = new ArrayList();
BL.Add(new BasisBlade(1 << 0, 1.0, "b_e1"));
BL.Add(new BasisBlade(1 << 1, 1.0, "b_e2"));
BL.Add(new BasisBlade(1 << 2, 1.0, "b_e3"));
Multivector B = new Multivector(BL);
System.Console.WriteLine("A = " + A.ToString());
System.Console.WriteLine("B = " + B.ToString());
System.Console.WriteLine("A B = " + Multivector.gp(A, B).ToString());
*/
/*
BasisBlade Z = new BasisBlade(1 << 0, 1.0, new Object[][] {
new Object[] { "blah", "aaaarg", 3.0, 2.0, 1 },
new Object[] { "bloooey", 'x', "crap", true, false},
new Object[] { "blah", "aaaarg", -10.0, 1 }
});
System.Console.WriteLine("Z = " + Z.ToString());
return;*/
// OK: now check for harder objects . . .
/*
Symbolic.HashtableSymbolicEvaluator HSE = new Symbolic.HashtableSymbolicEvaluator();
HSE.Add("a_e1", 1.0);
HSE.Add("a_e2", 2.0);
HSE.Add("a_e3", 3.0);
//Symbolic.RandomSymbolicEvaluator HSE = new Symbolic.RandomSymbolicEvaluator(-100.0, 100.0);
ArrayList AL = new ArrayList();
//AL.Add(new BasisBlade(1 << 0, 1.0, "a_no"));
AL.Add(new BasisBlade(1 << 1, 1.0, "a_e1"));
AL.Add(new BasisBlade(1 << 2, 1.0, "a_e2"));
AL.Add(new BasisBlade(1 << 3, 1.0, "a_e3"));
AL.Add(new BasisBlade(1 << 4, 1.0, "a_e1"));
Multivector A = new Multivector(AL);
A = Multivector.gp(A, Symbolic.ScalarOp.Exp(new Multivector(new BasisBlade(0, 1.0, "a_e2"))));
System.Console.WriteLine("A = " + A.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
// todo: check if exp, etc is working, then check why not collapse (by Multivector simplify, basis blade simplify) to single scalar value . .
Multivector Ae = A.SymbolicEval(HSE);
System.Console.WriteLine("Ae = " + Ae.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
return;
ArrayList BL = new ArrayList();
// BL.Add(new BasisBlade(1 << 0, 1.0));//, "b_no"));
BL.Add(new BasisBlade(1 << 1, -2.0));//, "b_e1"));
BL.Add(new BasisBlade(1 << 2, 3.0));//, "b_e2"));
BL.Add(new BasisBlade(1 << 3, -4.0));//, "b_e3"));
// BL.Add(new BasisBlade(1 << 4, 1.0));//, "b_ni"));
Multivector B = new Multivector(BL);
A = Multivector.op(A, B);
B = Multivector.gp(A, A);
System.Console.WriteLine("A = " + A.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("B = " + B.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
// how to know whether outcome of symbolic computation is positive, negative or zero???
// simply fill in values for random variables?
//Multivector unitA = A.Unit_r(M);
//System.Console.WriteLine("unitA = " + unitA.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
*/
/* Multivector ApAgB = Multivector.Add(Multivector.Add(A, Multivector.gp(A, B, M)), Multivector.gp(Multivector.GetPseudoscalar(4), 0.01));
// still testing . . .
System.Console.WriteLine("ApAgB = " + ApAgB.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
int g = ApAgB.LargestGradePartIndex();
System.Console.WriteLine("Largest grade part = " + g);
System.Console.WriteLine("Largest grade part = " + ApAgB.ExtractGrade(g).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("Top grade part = " + ApAgB.TopGradePartIndex(0.02));*/
/*
BasisBlade se1 = new BasisBlade(1 << 1, 1.0, "ae1");
BasisBlade se2 = new BasisBlade(1 << 2, 1.0, "ae2");
BasisBlade X = BasisBlade.op(se1, se2);
System.Console.WriteLine("se1 = " + se1);
System.Console.WriteLine("se2 = " + se2);
System.Console.WriteLine("X = " + X);
*/
/*
Multivector no = new Multivector(new BasisBlade(1, 1.0));
Multivector ni = new Multivector(new BasisBlade(1 << 4, 1.0));
Multivector noni = Multivector.GeometricProduct(no, ni, M);
System.Console.WriteLine("no = " + no.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("ni = " + ni.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("noni = " + noni.ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("noni2 = " + noni.ExtractGrade(2).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("dual(noni) = " + noni.Dual(M).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
System.Console.WriteLine("dual(I) = " + Multivector.GetPseudoscalar(5).Dual(M).ToString(new string[] { "no", "e1", "e2", "e3", "ni" }));
*/
/*BasisBlade no = new BasisBlade(1);
BasisBlade ni = new BasisBlade(1 << 4);
System.Console.WriteLine("no = " + no);
System.Console.WriteLine("ni = " + ni);
System.Collections.ArrayList L = BasisBlade.GeometricProduct(ni, no, M);
System.Console.WriteLine("L = " + BasisBlade.ToString(L, new string[] {"no", "e1", "e2", "e3", "ni"}));
*/
/*
BasisBlade A = new BasisBlade(1 | 4, 2.0);
BasisBlade B = new BasisBlade(2, 2.0);
BasisBlade C = new BasisBlade(1 | 4, 3.0);
BasisBlade D =new BasisBlade(1, 1.0);
System.Collections.ArrayList L = new System.Collections.ArrayList();
L.Add(A); L.Add(B); L.Add(C); L.Add(D); L.Add(new BasisBlade(2 | 4, 2.0));
System.Console.WriteLine("Before: ");
foreach (BasisBlade BB in L)
{
System.Console.WriteLine(BB.ToString());
}
L = BasisBlade.Simplify(L);
System.Console.WriteLine("After: ");
foreach (BasisBlade BB in L)
{
System.Console.WriteLine(BB.ToString());
}
*/
// System.Console.WriteLine("A = " + A);
// System.Console.WriteLine("B = " + B); // .ToString(new string[] { "no", "e1", "ni" })
// System.Console.WriteLine("A ^ B = " + (A ^ B));
/* try
{
double[][] MA = new double[][]{
new double[]{0.0, 0.0, 0.0, 0.0, -1.0},
new double[]{0.0, 1.0, 0.0, 0.0, 0.0},
new double[]{0.0, 0.0, 1.0, 0.0, 0.0},
new double[]{0.0, 0.0, 0.0, 1.0, 0.0},
new double[]{-1.0, 0.0, 0.0, 0.0, 0.0}};
new Metric(MA);
}
catch (Exception E)
{
System.Console.WriteLine(E.ToString());
}*/
/*
double[] A = new double[]{
1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
0.0, 0.0, 1.0};
int nbRows = 3;
int nbColumns = 3;
bool columnWise = false;
dnAnalytics.LinearAlgebra.Matrix M = dnAnalytics.LinearAlgebra.MatrixBuilder.CreateMatrix(A, nbRows, nbColumns, columnWise);
System.Console.WriteLine("Hi hoi!" + M);
*/
}
/// <summary>Computes the dual of <paramref name="A"/> with respect to the whole space. The
/// space has the dimension specified by the metric <paramref name="M"/>.</summary>
/// <param name="M">The metric to be used</param>
/// <returns>dual of this</returns>
public Multivector Dual(Metric M)
{
return Multivector.Dual(this, M);
}
