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A fast and easy to use dense matrix linear algebra library written in Java.
/*
* Copyright (c) 2009-2013, Peter Abeles. All Rights Reserved.
*
* This file is part of Efficient Java Matrix Library (EJML).
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.ejml.alg.block.decomposition.hessenberg;
import org.ejml.alg.block.BlockVectorOps;
import org.ejml.alg.block.decomposition.qr.BlockHouseHolder;
import org.ejml.data.D1Submatrix64F;
import org.ejml.ops.CommonOps;
import static org.ejml.alg.block.decomposition.qr.BlockHouseHolder.computeHouseHolderRow;
/**
* @author Peter Abeles
*/
public class TridiagonalBlockHelper {
/**
*
* Performs a tridiagonal decomposition on the upper row only.
*
*
*
* For each row 'a' in 'A':
* Compute 'u' the householder reflector.
* y(:) = A*u
* v(i) = y - (1/2)*(y^T*u)*u
* a(i+1) = a(i) - u*γ*v^T - v*u^t
*
*
* @param blockLength Size of a block
* @param A is the row block being decomposed. Modified.
* @param gammas Householder gammas.
* @param V Where computed 'v' are stored in a row block. Modified.
*/
public static void tridiagUpperRow( final int blockLength ,
final D1Submatrix64F A ,
final double gammas[] ,
final D1Submatrix64F V )
{
int blockHeight = Math.min(blockLength,A.row1-A.row0);
if( blockHeight <= 1 )
return;
int width = A.col1-A.col0;
int num = Math.min(width-1,blockHeight);
int applyIndex = Math.min(width,blockHeight);
// step through rows in the block
for( int i = 0; i < num; i++ ) {
// compute the new reflector and save it in a row in 'A'
computeHouseHolderRow(blockLength,A,gammas,i);
double gamma = gammas[A.row0+i];
// compute y
computeY(blockLength,A,V,i,gamma);
// compute v from y
computeRowOfV(blockLength,A,V,i,gamma);
// Apply the reflectors to the next row in 'A' only
if( i+1 < applyIndex ) {
applyReflectorsToRow( blockLength , A , V , i+1 );
}
}
}
/**
*
* Computes W from the householder reflectors stored in the columns of the row block
* submatrix Y.
*
*
*
* Y = v(1)
* W = -β1v(1)
* for j=2:r
* z = -β(I +WYT)v(j)
* W = [W z]
* Y = [Y v(j)]
* end
*
* where v(.) are the house holder vectors, and r is the block length. Note that
* Y already contains the householder vectors so it does not need to be modified.
*
*
*
* Y and W are assumed to have the same number of rows and columns.
*
*/
public static void computeW_row( final int blockLength ,
final D1Submatrix64F Y , final D1Submatrix64F W ,
final double beta[] , int betaIndex ) {
final int heightY = Y.row1-Y.row0;
CommonOps.fill(W.original, 0);
// W = -beta*v(1)
BlockHouseHolder.scale_row(blockLength,Y,W,0,1,-beta[betaIndex++]);
final int min = Math.min(heightY,W.col1-W.col0);
// set up rest of the rows
for( int i = 1; i < min; i++ ) {
// w=-beta*(I + W*Y^T)*u
double b = -beta[betaIndex++];
// w = w -beta*W*(Y^T*u)
for( int j = 0; j < i; j++ ) {
double yv = BlockHouseHolder.innerProdRow(blockLength,Y,i,Y,j,1);
BlockVectorOps.add_row(blockLength,W,i,1,W,j,b*yv,W,i,1,Y.col1-Y.col0);
}
//w=w -beta*u + stuff above
BlockHouseHolder.add_row(blockLength,Y,i,b,W,i,1,W,i,1,Y.col1-Y.col0);
}
}
/**
*
* Given an already computed tridiagonal decomposition, compute the V row block vector.
*
* y(:) = A*u
* v(i) = y - (1/2)*γ*(y^T*u)*u
*
*/
public static void computeV_blockVector( final int blockLength ,
final D1Submatrix64F A ,
final double gammas[] ,
final D1Submatrix64F V )
{
int blockHeight = Math.min(blockLength,A.row1-A.row0);
if( blockHeight <= 1 )
return;
int width = A.col1-A.col0;
int num = Math.min(width-1,blockHeight);
for( int i = 0; i < num; i++ ) {
double gamma = gammas[A.row0+i];
// compute y
computeY(blockLength,A,V,i,gamma);
// compute v from y
computeRowOfV(blockLength,A,V,i,gamma);
}
}
/**
*
* Applies the reflectors that have been computed previously to the specified row.
*
* A = A + u*v^T + v*u^T only along the specified row in A.
*
*
* @param blockLength
* @param A Contains the reflectors and the row being updated.
* @param V Contains previously computed 'v' vectors.
* @param row The row of 'A' that is to be updated.
*/
public static void applyReflectorsToRow( final int blockLength ,
final D1Submatrix64F A ,
final D1Submatrix64F V ,
int row )
{
int height = Math.min(blockLength, A.row1 - A.row0);
double dataA[] = A.original.data;
double dataV[] = V.original.data;
int indexU,indexV;
// for each previously computed reflector
for( int i = 0; i < row; i++ ) {
int width = Math.min(blockLength,A.col1 - A.col0);
indexU = A.original.numCols*A.row0 + height*A.col0 + i*width + row;
indexV = V.original.numCols*V.row0 + height*V.col0 + i*width + row;
double u_row = (i+1 == row) ? 1.0 : dataA[ indexU ];
double v_row = dataV[ indexV ];
// take in account the leading one
double before = A.get(i,i+1);
A.set(i,i+1,1);
// grab only the relevant row from A = A + u*v^T + v*u^T
BlockVectorOps.add_row(blockLength,A,row,1,V,i,u_row,A,row,row,A.col1-A.col0);
BlockVectorOps.add_row(blockLength,A,row,1,A,i,v_row,A,row,row,A.col1-A.col0);
A.set(i,i+1,before);
}
}
/**
*
* Computes the 'y' vector and stores the result in 'v'
*
* y = -γ(A + U*V^T + V*U^T)u
*
*
* @param blockLength
* @param A Contains the reflectors and the row being updated.
* @param V Contains previously computed 'v' vectors.
* @param row The row of 'A' that is to be updated.
*/
public static void computeY( final int blockLength ,
final D1Submatrix64F A ,
final D1Submatrix64F V ,
int row ,
double gamma )
{
// Elements in 'y' before 'row' are known to be zero and the element at 'row'
// is not used. Thus only elements after row and after are computed.
// y = A*u
multA_u(blockLength,A,V,row);
for( int i = 0; i < row; i++ ) {
// y = y + u_i*v_i^t*u + v_i*u_i^t*u
// v_i^t*u
double dot_v_u = BlockHouseHolder.innerProdRow(blockLength,A,row,V,i,1);
// u_i^t*u
double dot_u_u = BlockHouseHolder.innerProdRow(blockLength,A,row,A,i,1);
// y = y + u_i*(v_i^t*u)
// the ones in these 'u' are skipped over since the next submatrix of A
// is only updated
BlockVectorOps.add_row(blockLength,V,row,1,A,i,dot_v_u,V,row,row+1,A.col1-A.col0);
// y = y + v_i*(u_i^t*u)
// the 1 in U is taken account above
BlockVectorOps.add_row(blockLength,V,row,1,V,i,dot_u_u,V,row,row+1,A.col1-A.col0);
}
// y = -gamma*y
BlockVectorOps.scale_row(blockLength,V,row,-gamma,V,row,row+1,V.col1-V.col0);
}
/**
*
* Multiples the appropriate submatrix of A by the specified reflector and stores
* the result ('y') in V.
*
* y = A*u
*
*
* @param blockLength
* @param A Contains the 'A' matrix and 'u' vector.
* @param V Where resulting 'y' row vectors are stored.
* @param row row in matrix 'A' that 'u' vector and the row in 'V' that 'y' is stored in.
*/
public static void multA_u( final int blockLength ,
final D1Submatrix64F A ,
final D1Submatrix64F V ,
int row )
{
int heightMatA = A.row1-A.row0;
for( int i = row+1; i < heightMatA; i++ ) {
double val = innerProdRowSymm(blockLength,A,row,A,i,1);
V.set(row,i,val);
}
}
public static double innerProdRowSymm(int blockLength,
D1Submatrix64F A,
int rowA,
D1Submatrix64F B,
int rowB, int zeroOffset ) {
int offset = rowA + zeroOffset;
if( offset + B.col0 >= B.col1 )
return 0;
if( offset < rowB ) {
// take in account the one in 'A'
double total = B.get(offset,rowB);
total += BlockVectorOps.dot_row_col(blockLength,A,rowA,B,rowB,offset+1,rowB);
total += BlockVectorOps.dot_row(blockLength,A,rowA,B,rowB,rowB,A.col1-A.col0);
return total;
} else {
// take in account the one in 'A'
double total = B.get(rowB,offset);
total += BlockVectorOps.dot_row(blockLength,A,rowA,B,rowB,offset+1,A.col1-A.col0);
return total;
}
}
/**
*
* Final computation for a single row of 'v':
*
* v = y -(1/2)γ(y^T*u)*u
*
*
* @param blockLength
* @param A
* @param V
* @param row
* @param gamma
*/
public static void computeRowOfV( final int blockLength ,
final D1Submatrix64F A ,
final D1Submatrix64F V ,
int row ,
double gamma )
{
// val=(y^T*u)
double val = BlockHouseHolder.innerProdRow(blockLength,A,row,V,row,1);
// take in account the one
double before = A.get(row,row+1);
A.set(row,row+1,1);
// v = y - (1/2)gamma*val * u
BlockVectorOps.add_row(blockLength,V,row,1,A,row,-0.5*gamma*val,V,row,row+1,A.col1-A.col0);
A.set(row,row+1,before);
}
}