/// <summary>
/// Solve the minimum norm problem A * x = b => transpose(Q) * inv(L) * b. The minimum nrom problem is defined as:
/// from the infinite solutions of A * x = b, find the one with the minimum norm2(x). Warning: the rows of the original
/// matrix must be independent for this to work: If A (m-by-n, with m <= n) has full row rank, r = m => the column
/// space of A is an m dimensional subspace of R^m (every vector A*z is m-by-1) => the column space of A is the whole
/// R^m and A*x=b, always has at least one solution. The nullspace of A is an (n-m) dimensional subspace of R^n. Thus
/// there are infinite solutions to A*x=b.
/// </summary>
/// <param name="rhsVector">The right hand side vector b. It may lie outside the column space of the original matrix. Its
/// <see cref="IIndexable1D.Length"/> must be equal to this.<see cref="NumRows"/>.</param>
/// <exception cref="Exceptions.LapackException">Thrown if tha call to LAPACK fails due to <paramref name="rhsVector"/>
/// having a different <see cref="IIndexable1D.Length"/> than this.<see cref="NumRows"/>.</exception>
public Vector SolveMinNorm(Vector rhsVector) //TODO: perhaps I should use the driver routines of LAPACKE
{
if (NumRows > NumColumns)
{
throw new NotImplementedException("For now, the number of rows must be <= the number of columns");
}
Preconditions.CheckSystemSolutionDimensions(NumRows, rhsVector.Length);
// Min norm: A * x = b => L * Q * x = b, where b is the right hand side vector.
// Step 1: L * c = b. L is m-by-n, b is m-by-1 => c is n-by-1. Reminder n>=m.
// Decomposing L: [L1 0] * [c1; c2] = b => L1*c1 = b, where:
// L1 is m-by-m, lower triangular and stored in the factorized col major dat
// c1 is m-by-1 and can be found by forward substitution
// c2 is (n-m)-by-1 and can be any vector whatsoever.
int leadingDimA = NumRows; // Also k = min(NumRows, NumColumns) = NumRows = ldA
int incC = 1; // step in rhs array
double[] c = new double[NumColumns];
rhsVector.CopyToArray(0, c, 0, NumRows);
Blas.Dtrsv(StoredTriangle.Lower, TransposeMatrix.NoTranspose, DiagonalValues.NonUnit,
NumRows, reflectorsAndL, 0, leadingDimA, c, 0, incC);
// TODO: Check output of BLAS somehow. E.g. Zero diagonal entries will result in NaN in the result vector.
// Step 2: x = Q^T * c. Q is n-by-n, c is n-by-1, x is n-by-1
int numRowsC = NumColumns; // c has been grown past the limits of rhs.
int numColsC = 1; // c is m-by-1
int k = reflectorScalars.Length;
int leadingDimC = numRowsC;
LapackLeastSquares.Dormlq(MultiplicationSide.Left, TransposeMatrix.Transpose, numRowsC, numColsC, k,
reflectorsAndL, 0, leadingDimA, reflectorScalars, 0, c, 0, leadingDimC);
return(Vector.CreateFromArray(c, false));
}
/// <summary>
/// Solve the linear least squares problem A * x = b => x = inv(R) * transpose(Q) * b.
/// Warning: the columns of the original matrix A must be independent for this to work.
/// </summary>
/// <param name="rhsVector">The right hand side vector b. It may lie outside the column space of the original matrix. Its
/// <see cref="IIndexable1D.Length"/> must be equal to this.<see cref="NumRows"/>.</param>
/// <exception cref="Exceptions.LapackException">Thrown if the call to LAPACK fails due to <paramref name="rhsVector"/>
/// having a different <see cref="IIndexable1D.Length"/> than this.<see cref="NumRows"/>.</exception>
public Vector SolveLeastSquares(Vector rhsVector) //TODO: perhaps I should use the driver routines of LAPACKE
{
if (NumRows < NumColumns)
{
throw new NotImplementedException("For now, the number of rows must be >= the number of columns");
}
Preconditions.CheckSystemSolutionDimensions(NumRows, rhsVector.Length);
// Least squares: x = inv(A^T * A) * A^T * b = inv(R) * Q^T * b, where b is the right hand side vector.
// Step 1: c = Q^T * b. Q is m-by-m, b is m-by-1 => c is m-by-1
double[] c = rhsVector.CopyToArray();
int numRowsC = rhsVector.Length;
int numColsC = 1; // rhs = m-by-1
int numReflectors = reflectorScalars.Length;
int leadingDimA = numRowsC;
int leadingDimC = numRowsC;
LapackLeastSquares.Dormqr(MultiplicationSide.Left, TransposeMatrix.Transpose, numRowsC, numColsC, numReflectors,
reflectorsAndR, 0, leadingDimA, reflectorScalars, 0, c, 0, leadingDimC);
// Step 2: R * x = c, with R being m-by-n and upper trapezoidal (because m >= n).
// Decomposing R: [R1; 0] * x = [c1 ; c2 ] => R1 * x = c1 => R1 * x = c1, with R1 being n-by-n, upper triangular
// and stored in the factorized col major data. c1 and x are both n-by-1. The information stored in c2 is lost due to
// the least squares approximation.
// TODO: I do not really need to discard the extra m-n terms of c2, but I think it is unsafe to carry them around and
// risk some method of Vector using the length of the internal buffer, instead of its Length propert.
if (NumRows > NumColumns)
{
Array.Resize <double>(ref c, NumColumns);
}
int n = NumColumns; // Order of matrix R1
int ldR = NumRows; // R1 is stored in the upper trapezoid of a NumRows * NumColumns col major array.
int incC = 1; // step in rhs array, which is the same c used for Q^T * b
Blas.Dtrsv(StoredTriangle.Upper, TransposeMatrix.NoTranspose, DiagonalValues.NonUnit,
n, reflectorsAndR, 0, ldR, c, 0, incC);
// TODO: Check output of BLAS somehow. E.g. Zero diagonal entries will result in NaN in the result vector.
return(Vector.CreateFromArray(c, false));
}
/// <summary>
/// See https://software.intel.com/en-us/mkl-developer-reference-fortran-trsv#D8733073-F041-4AA1-B82C-123DFA993AD7
/// </summary>
public void Dtrsv(StoredTriangle uplo, TransposeMatrix transA, DiagonalValues diag, int n,
double[] a, int offsetA, int ldA, double[] x, int offsetX, int incX)
=> Blas.Dtrsv(uplo.Translate(), transA.Translate(), diag.Translate(), ref n, ref a[offsetA], ref ldA,
ref x[offsetX], ref incX);