The symmetric indefinite decomposition or Bunch-Kaufman decomposition
is defined for symmetric matrices that may not be positive definite.
It expresses a matrix as the product of a lower triangular matrix,
a block diagonal matrix, and the transpose of the triangular matrix.
The symmetric indefinite decomposition algorithm exploits the special structure
of symmetric matrices. As a result, it is about twice as fast as the LU decomposition.
Moreover, the decomposition is fairly stable, so it is suitable for use
in a broader range of problems.
Working with symmetric indefinite decompositions
The symmetric indefinite decomposition is implemented by the
SymmetricIndefiniteDecompositionT class.
It has no constructors. Instead, it is created by calling the
GetSymmetricIndefiniteDecomposition
method on the matrix. This method has two overloads. The first overload has no arguments.
The second overload takes a Boolean value
that specifies whether the contents of the matrix may be overwritten by the decomposition.
The default is .
var A = Matrix.CreateSymmetric(4, new double[] {
4.16,-3.12, 0.56,-0.10,
0 ,-5.03,-0.83, 1.18,
0 , 0 , 0.76, 0.34,
0 , 0 , 0 , 1.18
}, MatrixTriangle.Upper, MatrixElementOrder.RowMajor);
var ldlt = A.GetSymmetricIndefiniteDecomposition();Dim A = Matrix.CreateSymmetric(4, New Double() {
4.16, -3.12, 0.56, -0.1,
0, -5.03, -0.83, 1.18,
0, 0, 0.76, 0.34,
0, 0, 0, 1.18
}, MatrixTriangle.Upper, MatrixElementOrder.RowMajor)
Dim ldlt = A.GetSymmetricIndefiniteDecomposition()
No code example is currently available or this language may not be supported.
let A = Matrix.CreateSymmetric(4,
[|
4.16;-3.12; 0.56;-0.10;
0.0 ;-5.03;-0.83; 1.18;
0.0 ; 0.0 ; 0.76; 0.34;
0.0 ; 0.0 ; 0.0 ; 1.18
|], MatrixTriangle.Upper, MatrixElementOrder.RowMajor)
let ldlt = A.GetSymmetricIndefiniteDecomposition()
The Decompose method
performs the actual decomposition. This method copies the matrix if necessary. It then calls the appropriate LAPACK
routine to perform the actual decomposition.
This method is called by other methods as needed. You will rarely need to call it explicitly.
Once the decomposition is computed, a number of operations can be performed
in much less time. You can repeatedly solve a system of linear equations
with different right-hand sides. You can also calculate the
determinant and the inverse of the base matrix:
var b = Matrix.Create(4, 2, new double[] {
8.70,-13.35, 1.89,-4.14,
8.30, 2.13, 1.61, 5.00
}, MatrixElementOrder.ColumnMajor);
var x = ldlt.Solve(b);
var invA = ldlt.GetInverse();
var detA = ldlt.GetDeterminant();Dim b = Matrix.Create(4, 2, New Double() {
8.7, -13.35, 1.89, -4.14,
8.3, 2.13, 1.61, 5.0
}, MatrixElementOrder.ColumnMajor)
Dim x = ldlt.Solve(b)
Dim invA = ldlt.GetInverse()
Dim detA = ldlt.GetDeterminant()
No code example is currently available or this language may not be supported.
let b = Matrix.Create(4, 2,
[|
8.70;-13.35; 1.89;-4.14;
8.30; 2.13; 1.61; 5.00
|], MatrixElementOrder.ColumnMajor)
let x = ldlt.Solve(b)
let invA = ldlt.GetInverse()
let detA = ldlt.GetDeterminant()
The LowerTriangularFactor
property returns a TriangularMatrixT containing
the lower triangular matrix, G, of the decomposition.
There is no property to obtain the block diagonal matrix.
var L = ldlt.LowerTriangularFactor;
var LT = ldlt.UpperTriangularFactor;
Dim L = ldlt.LowerTriangularFactor
Dim LT = ldlt.UpperTriangularFactor
No code example is currently available or this language may not be supported.
let L = ldlt.LowerTriangularFactor
let LT = ldlt.UpperTriangularFactor