Conjugate Gradient

Collaboration diagram for Conjugate Gradient:

Warning

This MADlib method is still in early stage development. There may be some issues that will be addressed in a future version. Interface and implementation is subject to change.

About:

This function uses the iterative conjugate gradient method [1] to find a solution to the function:

\[ \boldsymbol Ax = \boldsymbol b \]

where \( \boldsymbol A \) is a symmetric, positive definite matrix and \(x\) and \( \boldsymbol b \) are vectors.

Input:

Matrix \( \boldsymbol A \) is assumed to be stored in a table where each row consists of at least two columns: array containing values of a given row, row number:

{TABLE|VIEW} matrix_A (
    row_number FLOAT,
    row_values FLOAT[],
)

The number of elements in each row should be the same.

\( \boldsymbol b \) is passed as a FLOAT[] to the function.

Usage:

Conjugate gradient can be called as follows:

SELECT conjugate_gradient('table_name', 
    'name_of_row_values_col', 'name_of_row_number_col', 'aray_of_b_values', 
    'desired_precision');

Function returns x as an array.

Examples:

1. Construct matrix A according to structure:

   sql> SELECT * FROM data;
    row_num | row_val 
   ---------+---------
          1 | {2,1}
          2 | {1,4}
   (2 rows)

2. Call conjugate gradient function:

   sql> SELECT conjugate_gradient('data','row_val','row_num','{2,1}',1E-6,1);
   INFO:  COMPUTE RESIDUAL ERROR 14.5655661859659
   INFO:  ERROR 0.144934004246004
   INFO:  ERROR 3.12963615962926e-31
   INFO:  TEST FINAL ERROR 2.90029642185163e-29
       conjugate_gradient     
   ---------------------------
    {1,-1.31838984174237e-15}
   (1 row)

Literature:

[1] "Conjugate gradient method" Wikipedia - http://en.wikipedia.org/wiki/Conjugate_gradient_method

See Also

File conjugate_gradient.sql_in documenting the SQL function.