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Matrix Decomposition Methods

The following methods are available for matrix decompositions and solution of simultaneous linear equations. 

1. T lud (valarray<size_t>& ri, T* pDet = NULL);
2. void lubksb (const valarray<size_t>& ri, const valarray<T>& v, valarray<T>& s) const;
3. void lumpove (const matrix<T>& ludm, const valarray<size_t>& ri, const valarray<T>& v, valarray<T>& s) const;
4. bool solve_lu (const valarray<T>& v, valarray<T>& s) const;

5. bool svd (matrix<T>& z, valarray<T>& w);
6. void svbksb (const matrix<T>& z, const valarray<T>& w, const valarray<T>& v, valarray<T>& s) const;
7. bool solve_sv (const valarray<T>& v, valarray<T>& s) const;
   
8. void qrd (matrix<T>& r);
9. void qrbksb (const matrix<T>& r, const valarray<T>& v, valarray<T>& s) const;
10. bool solve_qr (const valarray<T>& v, valarray<T>& s) const;

11. bool chold ();
12. void cholbksb (const valarray<T>& v, valarray<T>& s) const;
13. bool solve_chol (const valarray<T>& v, valarray<T>& s) const;

14. bool inv ();
15. bool inv_lu ();
16. bool inv_sv ();
17. bool inv_qr ();
   
18. bool solve (const valarray<T>& v, valarray<T>& s) const;
19. bool solve (const matrix<T>& v, matrix<T>& s) const;

  1. Performs LU decomposition on a square matrix, i.e., the matrix A is decomposed as,

    A = L . U

    Where L is a lower triangular matrix and U is an upper triangular matrix. On decomposition, the matrix A is replaced by both L and U. The new changed row index is returned in the valarray ri for performing the back substitution later. It can also optionally copy the determinant of the matrix in the variable pointed to by pDet. It returns true on successful decomposition, and false if the matrix is non-square or singular.


  2. Performs forward and back substitution on a LU decomposed matrix to find out the solution vector for a linear set of equations. Here, v is the right hand side input vector and ri is the row index as returned by lud method. The solution vector will be returned in the vector s.


  3. The lumpove method can be used to improve an imperfect solutions found by LU decomposition method earlier. This method must be applied to the original matrix. Here, ludm is the LU decomposition of the original matrix, v is the right hand side vector and ri is the row index. The improved solution will be returned in the vector s.


  4. The solve_lu method solves a set of linear equations using LU decomposition method. Here v is the right hand side vector, and s is the reference to a valarray object to receive the solution vector s. Returns true if successful; otherwise, it returns false.


  5. Performs Singular Value decomposition on a matrix, i.e., any M x N matrix A, where M >= N, is decomposed as,

    A = U . W . ZT

    Where U is an M x N column-orthogonal matrix, W is a diagonal matrix, and Z is an N x N orthogonal matrix. After successful decomposition, the original matrix A is replaced by U, the diagonal matrix is returned in the vector w, and the orthogonal matrix in z. Returns true on successful decomposition, and false if there is no convergence after 30 iterations.


  6. The svbksb method performs back substitution on a SV decomposed matrix to find out the solution vector for a linear set of equations. Here the matrix z and vector w are as returned by the svd method, v is the right hand side vector, and s is the solution vector to be returned by this method.


  7. The solve_sv method solves a set of linear equations using SV decomposition method. Here v is the right hand side vector, and s is the reference to a valarray object to receive the solution vector s. Returns true if successful; otherwise, it returns false.


  8. The QR decomposition method decomposes an M-by-N matrix A, where M >= N, into an M-by-N orthogonal matrix Q and an N-by-N upper triangular matrix R, i.e.,

    A = Q . R

    After successful decomposition, the original matrix A is replaced by the orthogonal matrix Q, and the upper triangular matrix R is returned in r.


  9. Finds the solution vector for a QR decomposed matrix. Here r is the upper triangular matrix returned by the qrd method, v is the right hand side vector, and s is the reference to a valarray class object to receive the solution vector.


  10. The solve_qr method solves a set of linear equations using QR decomposition method. Here v is the right hand side vector, and s is the reference to a valarray object to receive the solution vector s. Returns true if successful; otherwise, it returns false.


  11. The Cholesky decomposition method chold decomposes a symmetric, positive definite matrix A, into a lower triangular matrix L so that

    A = L . LT

    Returns true on success. If the matrix is not symmetric and positive definite, or singular, it returns false.


  12. The cholbksb method performs back substitution on a Cholesky decomposed matrix to find out the solution vector for a set of linear equations. Here v is the right hand side vector, and s is the reference to valarray object to receive the solution vector.


  13. The solve_chol solves a set of linear equations using Cholesky decomposition method. Here v is the right hand side vector, and s is the reference to a valarray object to receive the solution vector s. Returns true if successful; otherwise, it returns false.


  14. Inverts the matrix. The matrix is inverted in-place using minimum memory, i.e., the original matrix is replaced by its inverse.


  15. Inverts the matrix. The matrix is inverted using LU decomposition method. The original matrix is replaced by its inverse. This method uses more memory than inv and is also slightly slower.


  16. Inverts the matrix. The matrix is inverted using SV decomposition method. The original matrix is replaced by its inverse.


  17. Inverts the matrix. The matrix is inverted using QR decomposition method. The original matrix is replaced by its inverse.


  18. Solves a set of linear equations using LU decomposition method. Here v is the right hand side vector, and s is the reference to valarray object to receive the solution vector. Returns true is successful; otherwise, it returns false.


  19. Solves a set of linear equations using LU decomposition method. The matrix v may contain more than one the right hand side vector as column. For each right hand side vector, the solution vector is returned in the corresponding column of the matrix s. Returns true is successful; otherwise, it returns false.


Examples
typedef techsoft::matrix<double> Matrix;
typedef std::valarray<double> Vector;

using techsoft::gmslice;
using techsoft::LTRIANG;

Matrix A(5,5), Z(5,5);
Vector V(5), S(5), W(5);
std::valarray<size_t> ri;

std::cin >> A;          // Reads the matrix from stdin
std::cin >> Z[0];       // Reads the vector from stdin
V = Z[0];

A.lud( ri);
A.lubksb( ri, V, S);

A = B;
A.svd( Z, W);
A.svbksb( Z, W, V, S);

A = B;
A.qrd( Z);
A.qrbksb( Z, V, S);

A = B[gmslice(LTRIANG)];
A[gmslice(LTRIANG)] = ~A;       // A is now a symmetric matrix
A.chold();
A.cholbksb( V, S);

A = B;
A.inv();                        // Inverts the matrix

A = B;
A.solve( V, S);