282 lines
8.1 KiB
Java
282 lines
8.1 KiB
Java
import java.util.ArrayList;
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import java.util.List;
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public final class QRDecomposition {
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public static void main(String[] aArgs) {
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final double[][] data = new double [][] { { 12.0, -51.0, 4.0 },
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{ 6.0, 167.0, -68.0 },
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{ -4.0, 24.0, -41.0 },
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{ -1.0, 1.0, 0.0 },
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{ 2.0, 0.0, 3.0 } };
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// Task 1
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Matrix A = new Matrix(data);
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A.display("Initial matrix A:");
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MatrixPair pair = householder(A);
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Matrix Q = pair.q;
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Matrix R = pair.r;
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Q.display("Matrix Q:");
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R.display("Matrix R:");
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Matrix result = Q.multiply(R);
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result.display("Matrix Q * R:");
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// Task 2
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Matrix x = new Matrix(
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new double[][] { { 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0 } } );
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Matrix y = new Matrix(
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new double[][] { { 1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0 } } );
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result = fitPolynomial(x, y, 2);
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result.display("Result of fitting polynomial:");
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}
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private static MatrixPair householder(Matrix aMatrix) {
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final int rowCount = aMatrix.getRowCount();
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final int columnCount = aMatrix.getColumnCount();
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List<Matrix> versionsOfQ = new ArrayList<Matrix>(rowCount);
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Matrix z = new Matrix(aMatrix);
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Matrix z1 = new Matrix(rowCount, columnCount);
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for ( int k = 0; k < columnCount && k < rowCount - 1; k++ ) {
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Matrix vectorE = new Matrix(rowCount, 1);
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z1 = z.minor(k);
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Matrix vectorX = z1.column(k);
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double magnitudeX = vectorX.magnitude();
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if ( aMatrix.getEntry(k, k) > 0 ) {
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magnitudeX = -magnitudeX;
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}
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for ( int i = 0; i < vectorE.size(); i++ ) {
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vectorE.setEntry(i, 0, ( i == k ) ? 1 : 0);
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}
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vectorE = vectorE.scalarMultiply(magnitudeX).add(vectorX).unit();
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versionsOfQ.add(householderFactor(vectorE));
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z = versionsOfQ.get(k).multiply(z1);
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}
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Matrix Q = versionsOfQ.get(0);
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for ( int i = 1; i < columnCount && i < rowCount - 1; i++ ) {
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Q = versionsOfQ.get(i).multiply(Q);
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}
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Matrix R = Q.multiply(aMatrix);
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Q = Q.transpose();
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return new MatrixPair(R, Q);
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}
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public static Matrix householderFactor(Matrix aVector) {
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if ( aVector.getColumnCount() != 1 ) {
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throw new RuntimeException("Incompatible matrix dimensions.");
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}
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final int size = aVector.size();
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Matrix result = new Matrix(size, size);
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for ( int i = 0; i < size; i++ ) {
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for ( int j = 0; j < size; j++ ) {
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result.setEntry(i, j, -2 * aVector.getEntry(i, 0) * aVector.getEntry(j, 0));
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}
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}
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for ( int i = 0; i < size; i++ ) {
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result.setEntry(i, i, result.getEntry(i, i) + 1.0);
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}
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return result;
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}
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private static Matrix fitPolynomial(Matrix aX, Matrix aY, int aPolynomialDegree) {
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Matrix vandermonde = new Matrix(aX.getColumnCount(), aPolynomialDegree + 1);
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for ( int i = 0; i < aX.getColumnCount(); i++ ) {
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for ( int j = 0; j < aPolynomialDegree + 1; j++ ) {
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vandermonde.setEntry(i, j, Math.pow(aX.getEntry(0, i), j));
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}
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}
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return leastSquares(vandermonde, aY.transpose());
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}
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private static Matrix leastSquares(Matrix aVandermonde, Matrix aB) {
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MatrixPair pair = householder(aVandermonde);
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return solveUpperTriangular(pair.r, pair.q.transpose().multiply(aB));
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}
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private static Matrix solveUpperTriangular(Matrix aR, Matrix aB) {
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final int columnCount = aR.getColumnCount();
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Matrix result = new Matrix(columnCount, 1);
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for ( int k = columnCount - 1; k >= 0; k-- ) {
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double total = 0.0;
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for ( int j = k + 1; j < columnCount; j++ ) {
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total += aR.getEntry(k, j) * result.getEntry(j, 0);
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}
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result.setEntry(k, 0, ( aB.getEntry(k, 0) - total ) / aR.getEntry(k, k));
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}
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return result;
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}
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private static record MatrixPair(Matrix r, Matrix q) {}
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}
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final class Matrix {
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public Matrix(double[][] aData) {
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rowCount = aData.length;
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columnCount = aData[0].length;
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data = new double[rowCount][columnCount];
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for ( int i = 0; i < rowCount; i++ ) {
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for ( int j = 0; j < columnCount; j++ ) {
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data[i][j] = aData[i][j];
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}
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}
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}
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public Matrix(Matrix aMatrix) {
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this(aMatrix.data);
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}
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public Matrix(int aRowCount, int aColumnCount) {
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this( new double[aRowCount][aColumnCount] );
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}
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public Matrix add(Matrix aOther) {
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if ( aOther.rowCount != rowCount || aOther.columnCount != columnCount ) {
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throw new IllegalArgumentException("Incompatible matrix dimensions.");
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}
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Matrix result = new Matrix(data);
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for ( int i = 0; i < rowCount; i++ ) {
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for ( int j = 0; j < columnCount; j++ ) {
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result.data[i][j] = data[i][j] + aOther.data[i][j];
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}
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}
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return result;
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}
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public Matrix multiply(Matrix aOther) {
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if ( columnCount != aOther.rowCount ) {
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throw new IllegalArgumentException("Incompatible matrix dimensions.");
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}
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Matrix result = new Matrix(rowCount, aOther.columnCount);
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for ( int i = 0; i < rowCount; i++ ) {
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for ( int j = 0; j < aOther.columnCount; j++ ) {
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for ( int k = 0; k < rowCount; k++ ) {
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result.data[i][j] += data[i][k] * aOther.data[k][j];
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}
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}
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}
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return result;
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}
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public Matrix transpose() {
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Matrix result = new Matrix(columnCount, rowCount);
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for ( int i = 0; i < rowCount; i++ ) {
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for ( int j = 0; j < columnCount; j++ ) {
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result.data[j][i] = data[i][j];
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}
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}
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return result;
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}
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public Matrix minor(int aIndex) {
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Matrix result = new Matrix(rowCount, columnCount);
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for ( int i = 0; i < aIndex; i++ ) {
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result.setEntry(i, i, 1.0);
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}
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for ( int i = aIndex; i < rowCount; i++ ) {
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for ( int j = aIndex; j < columnCount; j++ ) {
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result.setEntry(i, j, data[i][j]);
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}
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}
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return result;
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}
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public Matrix column(int aIndex) {
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Matrix result = new Matrix(rowCount, 1);
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for ( int i = 0; i < rowCount; i++ ) {
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result.setEntry(i, 0, data[i][aIndex]);
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}
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return result;
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}
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public Matrix scalarMultiply(double aValue) {
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if ( columnCount != 1 ) {
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throw new IllegalArgumentException("Incompatible matrix dimension.");
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}
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Matrix result = new Matrix(rowCount, columnCount);
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for ( int i = 0; i < rowCount; i++ ) {
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result.data[i][0] = data[i][0] * aValue;
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}
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return result;
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}
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public Matrix unit() {
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if ( columnCount != 1 ) {
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throw new IllegalArgumentException("Incompatible matrix dimensions.");
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}
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final double magnitude = magnitude();
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Matrix result = new Matrix(rowCount, columnCount);
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for ( int i = 0; i < rowCount; i++ ) {
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result.data[i][0] = data[i][0] / magnitude;
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}
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return result;
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}
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public double magnitude() {
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if ( columnCount != 1 ) {
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throw new IllegalArgumentException("Incompatible matrix dimensions.");
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}
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double norm = 0.0;
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for ( int i = 0; i < data.length; i++ ) {
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norm += data[i][0] * data[i][0];
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}
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return Math.sqrt(norm);
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}
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public int size() {
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if ( columnCount != 1 ) {
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throw new IllegalArgumentException("Incompatible matrix dimensions.");
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}
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return rowCount;
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}
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public void display(String aTitle) {
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System.out.println(aTitle);
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for ( int i = 0; i < rowCount; i++ ) {
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for ( int j = 0; j < columnCount; j++ ) {
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System.out.print(String.format("%9.4f", data[i][j]));
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}
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System.out.println();
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}
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System.out.println();
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}
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public double getEntry(int aRow, int aColumn) {
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return data[aRow][aColumn];
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}
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public void setEntry(int aRow, int aColumn, double aValue) {
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data[aRow][aColumn] = aValue;
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}
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public int getRowCount() {
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return rowCount;
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}
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public int getColumnCount() {
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return columnCount;
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}
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private final int rowCount;
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private final int columnCount;
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private final double[][] data;
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}
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