/// <summary>
/// Decompose matrix A into uper and lower triangular part. Returns permutation matrix also.
/// </summary>
/// <param name="A">general input matrix. Size [m x n]</param>
/// <param name="U">[output] reference to upper triangular matrix. Size [min(m,n) x n]. Must not be null.</param>
/// <param name="P">[output] reference to permutation matrix. Size [min(m,n) x min(m,n)]. Must not be null.</param>
/// <returns>lower triangular matrix L of size [m x min(m,n)]</returns>
/// <remarks>A is decomposed into L and U, so that the equation
/// <code>ILMath.multiply(L,U) == ILMath.multiply(P,A)</code>
/// will hold except for round off error.
/// <para>L and U will be true lower triangular matrices.</para>
/// <example> <code>
/// //Let's construct a matrix X:
/// ILArray<double> X = new ILArray<double>(new double[]{1, 2, 3, 4, 4, 4, 5, 6, 7},3,3).T;
/// // now X.ToString() will give something like:
/// // {<Double> 63238509 [3x3] Ref(2)
/// //(:,:)
/// // 1,00000 2,00000 3,00000
/// // 4,00000 4,00000 4,00000
/// // 5,00000 6,00000 7,00000
/// //}
/// // construct references on U and P and call the decomposition
/// ILArray<double> U = new ILArray<double>.empty();
/// ILArray<double> P = new ILArray<double>.empty();
/// ILArray<double> L = ILMath.lu(X, ref U, ref P);
///
/// // L.ToString() is now:
/// // {<Double> 19634871 [3x3] Phys.
/// //(:,:)
/// // 1,00000 0,00000 0,00000
/// // 0,80000 1,00000 0,00000
/// // 0,20000 -1,00000 1,00000
/// //}
/// // U is now:
/// //{<Double> 22584602 [3x3] Phys.
/// //(:,:)
/// // 5,00000 6,00000 7,00000
/// // 0,00000 -0,80000 -1,60000
/// // 0,00000 0,00000 0,00000
/// //}
/// // and P is:
/// //{<Double> 2192437 [3x3] Phys.
/// //(:,:)
/// // 0,00000 0,00000 1,00000
/// // 0,00000 1,00000 0,00000
/// // 1,00000 0,00000 0,00000
/// //}
/// </code>
/// In order to reflect the pivoting done during the decomposition inside ?getrf, the matrix P may be used on A:
/// <code>
/// (ILMath.multiply(P,A) - ILMath.multiply(L,U)).ToString();
/// // will give:
/// //{<Double> 59192235 [3x3] Phys.
/// //(:,:)
/// // 0,00000 0,00000 0,00000
/// // 0,00000 0,00000 0,00000
/// // 0,00000 0,00000 0,00000
/// //}
/// </code>
/// </example>
/// <para>lu uses the Lapack function ?getrf.</para>
/// <para>All of the matrices U,L,P returned will be solid ILArrays.</para>
/// </remarks>
/// <seealso cref="ILNumerics.BuiltInFunctions.ILMath.lu(ILArray<double>)"/>
/// <seealso cref="ILNumerics.BuiltInFunctions.ILMath.lu(ILArray<double>, ref ILArray<double>)"/>
/// <exception cref="ILNumerics.Exceptions.ILArgumentException"> if input A is not a matrix.</exception>
public static ILArray <fcomplex> lu(ILArray <fcomplex> A, ref ILArray <fcomplex> U, ref ILArray <fcomplex> P)
{
if (!A.IsMatrix)
{
throw new ILArgumentSizeException("lu is defined for matrices only!");
}
int m = A.Dimensions[0], n = A.Dimensions[1], info = 0, minMN = (m < n)? m : n;
ILArray <fcomplex> L = (ILArray <fcomplex>)A.Clone();
int [] pivInd = ILMemoryPool.Pool.New <int>(minMN);
Lapack.cgetrf(m, n, L.m_data, m, pivInd, ref info);
if (info < 0)
{
ILMemoryPool.Pool.RegisterObject(pivInd);
throw new ILArgumentException("lu: illegal parameter error.");
//} else if (info > 0) {
// // singular diagonal entry found
}
else
{
// completed successfuly
if (!Object.Equals(U, null))
{
if (!Object.Equals(P, null))
{
pivInd = perm2indicesForward(pivInd);
U = copyUpperTriangle(L, minMN, n);
L = copyLowerTriangle(L, m, minMN, new fcomplex(1.0f, 0.0f));
P = ILArray <fcomplex> .zeros(minMN, minMN);
// construct permutation matrix P
for (int r = 0; r < m; r++)
{
P[r, pivInd[r]] = new fcomplex(1.0f, 0.0f);
}
}
else
{
pivInd = perm2indicesBackward(pivInd);
U = copyUpperTriangle(L, minMN, n);
L = copyLowerTrianglePerm(L, m, minMN, new fcomplex(1.0f, 0.0f), pivInd);
}
}
}
ILMemoryPool.Pool.RegisterObject(pivInd);
return(L);
}
/// <summary>
/// Determinant of square matrix
/// </summary>
/// <param name="A">square input matrix</param>
/// <returns>determinant of A</returns>
/// <remarks><para>The determinant is computed by decomposing A into upper and lower triangular part. Therefore LAPACK function ?getrf is used. <br />
/// Due to the properties of determinants, det(a) is the same as det(L) * det(U),where det(L) can easily be extracted from the permutation indices returned from LU decomposition. det(U) - with U beeing an upper triangular matrix - equals the product of the diagonal elements.</para>
/// <para>For scalar A, a plain copy of A is returned.</para></remarks>
/// <example>Creating a nonsingular 4x4 (double) matrix and it's determinant
/// <code>ILArray<double> A = ILMath.counter(1.0,1.0,4,4);
///A[1] = 0.0; // make A nonsingular
///A[14] = 0.0; //(same as: A[2,3] = 0.0;)
/// // A is now:
/// //<Double> [4,4]
/// //(:,:) 1e+001 *
/// // 0,10000 0,50000 0,90000 1,30000
/// // 0,00000 0,60000 1,00000 1,40000
/// // 0,30000 0,70000 1,10000 0,00000
/// // 0,40000 0,80000 1,20000 1,60000
///
///ILMath.det(A) gives:
/// //<Double> -360
///</code></example>
///<exception cref="ILNumerics.Exceptions.ILArgumentException">if A is empty or not a square matrix</exception>
public static ILArray <fcomplex> det(ILArray <fcomplex> A)
{
if (A.IsScalar)
{
return(A.C);
}
if (A.IsEmpty)
{
throw new ILArgumentException("det: A must be a matrix");
}
int m = A.Dimensions[0];
if (m != A.Dimensions[1])
{
throw new ILArgumentException("det: matrix A must be square");
}
ILArray <fcomplex> L = A.C;
int [] pivInd = new int[m];
int info = 0;
Lapack.cgetrf(m, m, L.m_data, m, pivInd, ref info);
if (info < 0)
{
throw new ILArgumentException("det: illegal parameter error.");
}
// determine pivoting: number of exchanges
fcomplex retA = new fcomplex(1.0f, 0.0f);
for (int i = 0; i < m;)
{
retA *= L.m_data[i * m + i];
if (pivInd[i] != ++i)
{
retA *= -1.0f;
}
}
L.Dispose();
return(new ILArray <fcomplex> (retA));
}
