LU Factorization of a Comples Sparse Matrix

Example: LU Factorization of a Complex Sparse Matrix

The LU Factorization of the sparse complex $6 \times 6$ matrix

$A=\begin{pmatrix} 10+7i & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 3+2i & -3+0i & -1+2i & 0.0 & 0.0 \\ 0.0 & 0.0 & 4+2i & 0.0 & 0.0 & 0.0 \\ -2-4i & 0.0 & 0.0 & 1+6i & -1+3i & 0.0 \\ -5+4i & 0.0 & 0.0 & -5+0i & 12+2i & -7+7i \\ -1+12i & -2+8i & 0.0 & 0.0 & 0.0 & 3+7i \end{pmatrix}$ is computed. The sparse coordinate form for A is given by row, column, value triplets:

row	column	value
0	0	10+7i
1	1	3+2i
1	2	-3+0i
1	3	-1+2i
2	2	4+2i
3	0	-2-4i
3	3	1+6i
3	4	-1+3i
4	0	-5+4i
4	3	-5+0i
4	4	12+2i
4	5	-7+7i
5	0	-1+12i
5	1	-2+8i
5	5	3+7i

Let

so that $b_1:=Ax = {(3+17i, -19+5i, 6+18i, -38+32i, -63+49i, -57+83i)}^T$ and $b_2:=A^Hx = {(54-112i, 46-58i, 12, 5-51i, 78+34i, 60-94i)}^T\,.$

The LU factorization of is used to solve the complex sparse linear systems and with iterative refinement. The reciprocal pivot growth factor and the reciprocal condition number are also computed.

using System;
using Imsl.Math;

public class ComplexSuperLUEx1
{    
    public static void Main(String[] args)
    {
        int m;
        ComplexSuperLU cslu;
        double conditionNumber, recip_pivot_growth;
        Complex[] sol = null;
        double Ferr, Berr;
        Complex[] b1 = { new Complex(3.0, 17.0), new Complex(-19.0, 5.0),
                         new Complex(6.0, 18.0), new Complex(-38.0, 32.0),
                         new Complex(-63.0, 49.0), new Complex(-57.0, 83.0)
                       };
        Complex[] b2 = { new Complex(54.0, -112.0), new Complex(46.0, -58.0),
                         new Complex(12.0, 0.0), new Complex(5.0, -51.0),
                         new Complex(78.0, 34.0), new Complex(60.0, -94.0)
                       };
        // Initialize input matrix A.
        m = 6;
        ComplexSparseMatrix a = new ComplexSparseMatrix(m, m);
        a.Set(0, 0, new Complex(10.0, 7.0));
        a.Set(1, 1, new Complex(3.0, 2.0));
        a.Set(1, 2, new Complex(-3.0, 0.0));
        a.Set(1, 3, new Complex(-1.0, 2.0));
        a.Set(2, 2, new Complex(4.0, 2.0));
        a.Set(3, 0, new Complex(- 2.0, -4.0));
        a.Set(3, 3, new Complex(1.0, 6.0));
        a.Set(3, 4, new Complex(-1.0, 3.0));
        a.Set(4, 0, new Complex(-5.0, 4.0));
        a.Set(4, 3, new Complex(-5.0, 0.0));
        a.Set(4, 4, new Complex(12.0, 2.0));
        a.Set(4, 5, new Complex(-7.0, 7.0));
        a.Set(5, 0, new Complex(-1.0, 12.0));
        a.Set(5, 1, new Complex(-2.0, 8.0));
        a.Set(5, 5, new Complex(3.0, 7.0));

        // Compute the sparse LU factorization of a
        cslu = new ComplexSuperLU(a);
        cslu.Equilibrate = false;
        cslu.ColumnOrderingMethod =
                ComplexSuperLU.ColumnOrdering.Natural;
        cslu.PivotGrowth =true;

        // Set option of iterative refinement
        cslu.IterativeRefinement=true;

        // Solve sparse system A*x = b1
        Console.Out.WriteLine();
        Console.Out.WriteLine("Solve sparse System A*x=b1");
        Console.Out.WriteLine("==========================");
        Console.Out.WriteLine();
        sol = cslu.Solve(b1);
        new PrintMatrix("Solution").Print(sol);

        // Determine error bounds
        Ferr = cslu.ForwardErrorBound;
        Berr = cslu.RelativeBackwardError;
        Console.Out.WriteLine();
        Console.Out.WriteLine("Forward error bound: " + Ferr);
        Console.Out.WriteLine("Relative backward error: " + Berr);
        Console.Out.WriteLine();
        Console.Out.WriteLine();

        // Solve sparse system (A^H)*x = b2
        Console.Out.WriteLine();
        Console.Out.WriteLine("Solve sparse System (A^H)*x=b2");
        Console.Out.WriteLine("==============================");
        Console.Out.WriteLine();
        sol = cslu.SolveConjugateTranspose(b2);
        new PrintMatrix("Solution").Print(sol);

        // Determine error bounds
        Ferr = cslu.ForwardErrorBound;
        Berr = cslu.RelativeBackwardError;
        Console.Out.WriteLine();
        Console.Out.WriteLine("Forward error bound: " + Ferr);
        Console.Out.WriteLine("Relative backward error: " + Berr);
        Console.Out.WriteLine();
        Console.Out.WriteLine();

        // Compute reciprocal pivot growth factor and condition number
        recip_pivot_growth = cslu.ReciprocalPivotGrowthFactor;
        conditionNumber = cslu.ConditionNumber;
        Console.Out.WriteLine("Pivot growth factor and condition number");
        Console.Out.WriteLine("========================================");
        Console.Out.WriteLine();
        Console.Out.WriteLine("Reciprocal pivot growth factor: " + recip_pivot_growth);
        Console.Out.WriteLine("Reciprocal condition number: " + conditionNumber);
        Console.Out.WriteLine();
    }
}

Output


Solve sparse System A*x=b1
==========================

Solution
    0    
0  1+1i  
1  2+2i  
2  3+3i  
3  4+4i  
4  5+5i  
5  6+6i  


Forward error bound: 2.83933305928053E-15
Relative backward error: 1.70803542250024E-16



Solve sparse System (A^H)*x=b2
==============================

        Solution
            0           
0  1+1i                 
1  2+2.00000000000001i  
2  3+3i                 
3  4+4i                 
4  5+5i                 
5  6+6i                 


Forward error bound: 8.54834098797111E-15
Relative backward error: 1.02977208081174E-16


Pivot growth factor and condition number
========================================

Reciprocal pivot growth factor: 0.799382716049383
Reciprocal condition number: 0.0700654479096751

Link to C# source.