Does anyone know how to interpret the permutation vector in GMatrix.LUD()?
I have passed it blindly into GVector.LUDBackSolve() and it appears to work
fine, but I would like to know how to form the permutation matrix P from the
permutation vector generated by GMatrix.LUD().
For example, how does the permutation vector {1.0 1.0 2.0 } imply the
following permutation matrix?
0 1 0
1 0 0
0 0 1
(see full example below)
Thanks much.
Sincerely,
Dave
=======================================
Here is an example...(I don't know how to interpret the { 1, 1, 2 }
permutation vector. My understanding of a permutation matrix is that it is
an identity matrix with interchanged rows. I understand that the
permutation vector encodes this row interchange, but I can't figure out how.
original matrix (A):
0.0 -2.0 2.0
1.0 2.0 -1.0
3.0 5.0 -8.0
count=-1
LUD matrix as output from GMatrix.LUD():
1.0 2.0 -1.0
0.0 -2.0 2.0
3.0 0.5 -6.0
LUD matrix (L | U):
1 0 0 | 1.0 2.0 -1.0
0.0 1 0 | 0 -2.0 2.0
3.0 0.5 1 | 0 0 -6.0
Permutation vector:
1.0 1.0 2.0
Since I know that PA = LU and LU =
1.0 2.0 -1.0
0.0 -2.0 2.0
3.0 5.0 -8.0
I can infer that P =
0 1 0
1 0 0
0 0 1
(which is correct, since this is the permutation matrix that permutes A into
LU).
But, how the heck does a permutation vector { 1, 1, 2 } imply the
aforementioned permutation matrix?
FYI, this is the same matrix used at
http://www.cs.ut.ee/~toomas_l/linalg/lin2/node4.html#permutationmatrix for
those who want to see the math behind this example.
=======================================
The code below was used to generate the previous output.
void jMenuItemGMatrixTest_actionPerformed(ActionEvent e)
{
final int SIZE = 3;
GMatrix matrix = new GMatrix(SIZE, SIZE);
matrix.setElement(0, 0, 0); // see Note2 above
matrix.setElement(0, 1, -2);
matrix.setElement(0, 2, 2);
matrix.setElement(1, 0, 1);
matrix.setElement(1, 1, 2);
matrix.setElement(1, 2, -1);
matrix.setElement(2, 0, 3);
matrix.setElement(2, 1, 5);
matrix.setElement(2, 2, -8);
System.out.println("original matrix:");
trace(matrix);
GMatrix LUD = new GMatrix(SIZE,SIZE);
GVector permutation = new GVector(SIZE); // see Note1 above
int count = matrix.LUD(LUD, permutation);
System.out.println("count=" + count);
System.out.println("LUD matrix:");
trace(LUD);
System.out.println("LUD matrix:");
traceLUD(LUD);
System.out.println("Permutation vector:");
J3DOps.trace(permutation);
} // end jMenuItemGMatrixTest_actionPerformed()
=======================================
The following description explains what a permutation matrix is (source:
http://www.cs.rochester.edu/u/www/u/pawlicki/PROJECT2.htm).
Permutation Matrix
A permutation matrix is a square matrix with exactly one 1 in each row and
column and zeros elsewhere (i.e. and identity matrix with its rows and
columns permuted). Multiplying a permutation matrix P, by any matrix A will
yield the same A matrix, with rows swapped.
Consider multiplying matrixP x matrixA if defined as follows:
double matrixA[][] = {{1,2,3,4},{4,5,6,7},{7,8,9,10}};
double matrixP[][] = {{1,0,0},{0,0,1},{0,1,0}};
It can be shown that the resultant matrix C will be
{{1,2,3,4},{7,8,9,10},{4,5,6,7}};
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