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Symbolic LDL and Cholesky Factorizations

Symbolic LDL and Cholesky factorization algorithms are meant for symmetric positive definite systems; hence, again, only the lower-triangular part of the matrix must be provided. The PROC IML function SOLVELIN provides an interface to both algorithms; the minimum degree ordering heuristic is invoked automatically as follows:

  
    SOLVELIN (x, status, A, b, method)
 

x

solution vector

status

status indicator 0 success, 1 matrix is not positive-definite, 2 out of memory

A

sparse matrix (lower-triangular part)

b

vector of right-hand sides

method

a character string, which specifies factorization type, possible values: "LDL" for LDL factorization, and "CHOL" for Cholesky.

The code for this example is as follows:

  
  
    /* value   row   col */ 
    A = { 3     1     1, 
          1     2     1, 
          4     2     2, 
          1     3     2, 
          3     4     2, 
         10     3     3, 
          3     4     4 }; 
  
    /* right-hand side */ 
    b = {1, 1, 1, 1}; 
  
    /* invoke LDL factorization */ 
    call solvelin (x, status, a, b, "LDL"); 
  
    print x; /* print solution */
 

Here is the program output:

  
                     X 
                 0.5882353 
                 -0.764706 
                 0.1764706 
                 1.0980392