com.numericalmethod.suanshu.matrix.doubles.factorization.qr
Class GramSchmidt

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.factorization.qr.GramSchmidt

All Implemented Interfaces:

QRDecomposition

public class GramSchmidt
extends java.lang.Object
implements QRDecomposition

The Gram–Schmidt process is a method for orthogonalizing a set of vectors in an inner product space. It does so by iteratively computing the vector orthogonal to the subspace spanned by previously found orthogonal vectors.

The orthogonal vector is the difference between a column vector and its projection on the subspace.

There is the problem of "loss of orthogonality" during the process, which gives arise to the precision error or rounding error. In general, the bigger the matrix is, e.g., dimension = 3500x3500, the less precise the results are. Also, the vectors in Q may not be as orthogonal.

A numerically stable Gram–Schmidt process with twice re-orthogonalization is implemented to alleviate the problem of rounding errors.

Numerical determination of rank requires a criterion to decide when a value should be treated as zero. This is a practical choice which depends on both the matrix and the application. For instance, for a matrix with a big first eigenvector, we should accordingly decrease the precision to compute the rank.

While the results for the orthogonal basis may match those of the Householder Reflection HouseholderReflection, the results for the orthogonal complement may differ because the kernel basis is not unique.

See Also:

• On the loss of orthogonality in the Gram-Schmidt orthogonalization process. Luc Giraud, Julien Langou, Miroslav Rozloznik. Computers & Mathematics with Applications. Volume 50, Issue 7, October 2005, p. 1069-1075. Numerical Methods and Computational Mechanics.
• Matrix Computations (Johns Hopkins Studies in Mathematical Sciences)(3rd Edition)(Paperback). Gene H. Golub, Charles F. Van LoanGene H. Golub, Charles F. Van Loan
• Wikipedia: Gram–Schmidt process

Field Summary

double	epsilon a precision parameter: when a number |x| ≤ ε, it is considered 0
int	ncols number of columns
int	nrows number of rows

Constructor Summary

GramSchmidt(Matrix A)
Construct an instance of the Gram-Schmidt process to orthogonalize a matrix.

GramSchmidt(Matrix A, boolean pad0Cols, double epsilon)
Construct an instance of the Gram-Schmidt process to orthogonalize a matrix.

Method Summary

PermutationMatrix	P() Get a copy of P, the pivoting matrix in the QR decomposition.
Matrix	Q() Get a copy of the orthogonal Q matrix in the QR decomposition.
UpperTriangularMatrix	R() Get a copy of the upper triangular matrix R in the QR decomposition.
int	rank() Get the numerical rank of the matrix A as computed by the QR decomposition.
Matrix	squareQ() Get a copy of the square Q matrix.
Matrix	tallR() Get a copy of the tall R matrix.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

nrows

public final int nrows

number of rows

ncols

public final int ncols

number of columns

epsilon

public final double epsilon

a precision parameter: when a number |x| ≤ ε, it is considered 0

Constructor Detail

GramSchmidt

public GramSchmidt(Matrix A,
                   boolean pad0Cols,
                   double epsilon)

Construct an instance of the Gram-Schmidt process to orthogonalize a matrix.

Parameters:

A - the matrix to orthogonalize

pad0Cols - when a column is linearly dependent on the previous columns, there is no orthogonal vector. We pad the basis with a 0-vector.

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

GramSchmidt

public GramSchmidt(Matrix A)

Construct an instance of the Gram-Schmidt process to orthogonalize a matrix.

Parameters:

A - a matrix

Method Detail

Q

public Matrix Q()

Description copied from interface: QRDecomposition

Get a copy of the orthogonal Q matrix in the QR decomposition.

 A = QR
 

Dimension of Q is nrows x ncols, same as A, the matrix to orthogonalize.

Specified by:

Q in interface QRDecomposition

Returns:

a copy of the Q matrix in the QR decomposition

R

public UpperTriangularMatrix R()

Description copied from interface: QRDecomposition

Get a copy of the upper triangular matrix R in the QR decomposition.

 A = QR
 

Dimension of R is ncols x ncols, a square matrix.

Specified by:

R in interface QRDecomposition

Returns:

a copy of the upper triangular matrix R in the QR decomposition

P

public PermutationMatrix P()

Description copied from interface: QRDecomposition

Get a copy of P, the pivoting matrix in the QR decomposition.

Specified by:

P in interface QRDecomposition

Returns:

a copy of the P pivoting matrix in the QR decomposition

rank

public int rank()

Description copied from interface: QRDecomposition

Get the numerical rank of the matrix A as computed by the QR decomposition.

Numerical determination of rank requires a criterion to decide when a value should be treated as zero.

This is a practical choice which depends on both the matrix and the application. For instance, for a matrix with a big first eigenvector, we should accordingly decrease the precision to compute the rank.

You may need to change the precision parameter to accurately compute the rank. See the test cases for example.

Specified by:

rank in interface QRDecomposition

Returns:

the rank of A

squareQ

public Matrix squareQ()

Get a copy of the square Q matrix. This is an arbitrary orthogonal completion of the Q matrix in the QR decomposition. Dimension is nrows x nrows (square).

 A = square_Q %*% tall_R
 

In this implementation, we extend Q by adding A's orthogonal complement at the end. Suppose Q has the orthogonal basis for a subspace A. To compute the orthogonal complement of A, we can apply the Gram-Schmidt procedure to either

1. the columns of I - P = I - Q %*% Q.t(), or
2. the spanning set {u1, u2, ... uk, e1, e2, ... en} and keeping only the first n elements of the resulting basis of Rn. The last n-k elements are the basis for the orthogonal complement. k is the rank.

We implemented the second option.

Specified by:

squareQ in interface QRDecomposition

Returns:

the spanning set of both the orthogonal basis of A and the orthogonal complement

tallR

public Matrix tallR()

Description copied from interface: QRDecomposition

Get a copy of the tall R matrix. This is completed by binding zero rows beneath the square upper triangular matrix R in the QR decomposition. Dimension is nrows x ncols. Note that this may no longer be square.

 A = square_Q %*% tall_R
 

Specified by:

tallR in interface QRDecomposition

Returns:

a copy of the tall R matrix