Package com.imsl.math

Class SparseCholesky

java.lang.Object
com.imsl.math.SparseCholesky
All Implemented Interfaces:
Serializable
public class SparseCholesky extends Object implements Serializable
Sparse Cholesky factorization of a matrix of type SparseMatrix.

Class SparseCholesky computes the Cholesky factorization of a sparse symmetric positive definite matrix A. This factorization can then be used to compute the solution of the linear system \(Ax = b\).

Typically, the solution of a large sparse positive definite system \(Ax = b\) is done in 4 steps:

  1. In step one, an ordering algorithm is used to preserve sparsity in the Cholesky factor L of matrix A during the numerical factorization process. The new order can be described by a permutation matrix P.
  2. Step two consists of setting up the data structure for the Cholesky factor L, where \(PAP^T=LL^T\). This step is called the symbolic factorization phase of the computation. During symbolic factorization, only the sparsity pattern of sparse matrix A, i.e., the locations of the nonzero entries of matrix A are needed but not any of the elements themselves.
  3. In step 3, the numerical factorization phase, the Cholesky factorization is done numerically.
  4. Step 4 is the solution phase. Here, the numerical solution, x, to the original system is obtained by solving the two triangular systems \(Ly_1=Pb\), \(L^Ty_2=y_1\) and the permutation \(x=P^Ty_2\).

Class SparseCholesky realizes all four steps by algorithms described in George and Liu (1981). Especially, step one, is a realization of a minimum degree ordering algorithm. The numerical factorization in its standard form is based on a sparse compressed storage scheme. Alternatively, a multifrontal method can be used. The multifrontal method requires more storage but will be faster than the standard method in certain cases. The multifrontal method is based on the routines in Liu (1987). For a detailed description of this method, see Liu (1990), also Duff and Reid(1983, 1984), Ashcraft (1987) et al. (1987), and Liu (1986, 1989, 1992). The numerical factorization method can be specified by using the setNumericFactorizationMethod.

The solve method will compute the symbolic and numeric factorizations if they have not already been computed or supplied by the user through the factorSymbolically, factorNumerically, setNumericFactor, or setSymbolicFactor methods. These factorizations are retained for later use by the solve method when different right-hand sides are to be solved.

There is a special situation where computations can be simplified. If an application generates different sparse symmetric positive definite coefficient matrices that all have the same sparsity pattern, then by using methods getSymbolicFactor and setSymbolicFactor the symbolic factorization needs only be computed once.

See Also:

  • Field Details

    • STANDARD_METHOD

      public static final int STANDARD_METHOD
      Indicates the method of George/Liu (1981) will be used for numeric factorization.
      See Also:

    • MULTIFRONTAL_METHOD

      public static final int MULTIFRONTAL_METHOD
      Indicates the multifrontal method will be used for numeric factorization.
      See Also:

  • Constructor Details

    • SparseCholesky

      public SparseCholesky(SparseMatrix A)
      Constructs the matrix structure for the Cholesky factorization of a sparse symmetric positive definite matrix of type SparseMatrix.
      Parameters:
      A - The SparseMatrix symmetric positive definite matrix to be factored. Only the lower triangular part of the input matrix is used.

  • Method Details

    • solve

      public double[] solve(double[] b) throws SparseCholesky.NotSPDException
      Computes the solution of a sparse real symmetric positive definite system of linear equations \(Ax=b\).

      This method solves the linear system \(Ax=b\), where A is symmetric positive definite. The solution is obtained in several steps:

      1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite system \(PAP^T=Pb\).
      2. Then, matrix \(PAP^T\) is symbolically and numerically factored.
      3. The final solution is obtained by solving the systems \(Ly_1=Pb, L^Ty_2=y_1\) and \(x=P^Ty_2\).

      By default this method implements all of the above steps. The factorizations are retained for later use by subsequent solves. By choosing appropriate methods within this class, the computation can be reduced to the solution of the system \(Ax=b\) for a given or precomputed symbolic or numeric factor.

      Parameters:
      b - a double vector of length equal to the order of matrix A representing the new right-hand side.
      Returns:
      a double vector of length equal to the order of matrix A representing the solution to the system of linear equations \(Ax=b\).
      Throws:
      SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.

    • factorNumerically

      public void factorNumerically() throws SparseCholesky.NotSPDException
      Computes the numeric factorization of a sparse real symmetric positive definite matrix.

      This method numerically factors the instance of the constructed matrix A, where A is of type SparseMatrix and is symmetric positive definite. The factorization is obtained in several steps:

      1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite matrix \(PAP^T\).
      2. Then, matrix \(PAP^T\) is symbolically and numerically factored.

      Note that the symbolic factorization is not done if the symbolic factor has been supplied by the user through the setSymbolicFactor method.

      Throws:
      SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.

    • factorSymbolically

      public void factorSymbolically() throws SparseCholesky.NotSPDException
      Computes the symbolic factorization of a sparse real symmetric positive definite matrix.

      This method symbolically factors the instance of the constructed matrix A, where A is of type SparseMatrix and is symmetric positive definite. The factorization is obtained in several steps:

      1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite system \(PAP^T=Pb\).
      2. Then, matrix \(PAP^T\) is symbolically factored.

      Throws:
      SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.

    • setSymbolicFactor

      public void setSymbolicFactor(SparseCholesky.SymbolicFactor symbolicFactor)
      Sets the symbolic Cholesky factor to use in solving a sparse positive definite system of linear equations \(Ax=b\).
      Parameters:
      symbolicFactor - a SymbolicFactor containing the symbolic Cholesky factor. By default the symbolic factorization is computed.

    • getSymbolicFactor

      public SparseCholesky.SymbolicFactor getSymbolicFactor()
      Returns the symbolic Cholesky factor.
      Returns:
      a SymbolicFactor containing the symbolic Cholesky factor.

    • setNumericFactor

      public void setNumericFactor(SparseCholesky.NumericFactor numericFactor)
      Sets the numeric Cholesky factor to use in solving of a sparse positive definite system of linear equations \(Ax=b\).
      Parameters:
      numericFactor - a NumericFactor object containing the numeric Cholesky factor. By default the numeric factorization is computed.

    • getNumericFactor

      public SparseCholesky.NumericFactor getNumericFactor()
      Returns the numeric Cholesky factor.
      Returns:
      a NumericFactor containing the numeric Cholesky factor.

    • setNumericFactorizationMethod

      public void setNumericFactorizationMethod(int method)
      Defines the method used in the numerical factorization of the permuted input matrix.
      Parameters:
      method - an int value specifying the method to choose:
      Method Name Description
      STANDARD_METHOD standard method as described by George/Liu (1981). This is the default.
      MULTIFRONTAL_METHOD multifrontal method
      Throws:
      IllegalArgumentException - This exception is thrown when the value for method is not STANDARD_METHOD or MULTIFRONTAL_METHOD.

    • getNumericFactorizationMethod

      public int getNumericFactorizationMethod()
      Returns the method used in the numerical factorization of the permuted input matrix.
      Returns:
      an int value equal to STANDARD_METHOD = 0 or MULTIFRONTAL_METHOD = 1 representing the method used in the numeric factorization of the permuted input matrix. See setNumericFactorizationMethod for more details.

    • getSmallestDiagonalElement

      public double getSmallestDiagonalElement()
      Returns the smallest diagonal element of the Cholesky factor.
      Returns:
      a double value specifying the smallest diagonal element of the Cholesky factor. Use of this method is only sensible if a numeric factorization of the input matrix was done beforehand.

    • getLargestDiagonalElement

      public double getLargestDiagonalElement()
      Returns the largest diagonal element of the Cholesky factor.
      Returns:
      a double value specifying the largest diagonal element of the Cholesky factor. Use of this method is only sensible if a numeric factorization of the input matrix was done beforehand.

    • getNumberOfNonzeros

      public long getNumberOfNonzeros()
      Returns the number of nonzeros in the Cholesky factor.
      Returns:
      a long value specifying the number of nonzeros (including the diagonal) of the Cholesky factor.