矩陣計算

package Jama;import java.text.NumberFormat; import java.text.DecimalFormat; import java.text.DecimalFormatSymbols; import java.util.Locale; import java.text.FieldPosition; import java.io.PrintWriter; import java.io.BufferedReader; import java.io.StreamTokenizer; import Jama.util.*;/**Jama = Java Matrix class. <P>The Java Matrix Class provides the fundamental operations of numericallinear algebra. Various constructors create Matrices from two dimensionalarrays of double precision floating point numbers. Various "gets" and"sets" provide access to submatrices and matrix elements. Several methods implement basic matrix arithmetic, including matrix addition andmultiplication, matrix norms, and element-by-element array operations.Methods for reading and printing matrices are also included. All theoperations in this version of the Matrix Class involve real matrices.Complex matrices may be handled in a future version. <P>Five fundamental matrix decompositions, which consist of pairs or triplesof matrices, permutation vectors, and the like, produce results in fivedecomposition classes. These decompositions are accessed by the Matrixclass to compute solutions of simultaneous linear equations, determinants,inverses and other matrix functions. The five decompositions are: <P><UL><LI>Cholesky Decomposition of symmetric, positive definite matrices.<LI>LU Decomposition of rectangular matrices.<LI>QR Decomposition of rectangular matrices.<LI>Singular Value Decomposition of rectangular matrices.<LI>Eigenvalue Decomposition of both symmetric and nonsymmetric square matrices. </UL> <DL> <DT><B>Example of use:</B></DT> <P> <DD>Solve a linear system A x = b and compute the residual norm, ||b - A x||. <P><PRE>double[][] vals = {{1.,2.,3},{4.,5.,6.},{7.,8.,10.}};Matrix A = new Matrix(vals);Matrix b = Matrix.random(3,1);Matrix x = A.solve(b);Matrix r = A.times(x).minus(b);double rnorm = r.normInf(); </PRE></DD> </DL>@author The MathWorks, Inc. and the National Institute of Standards and Technology. @version 5 August 1998 */public class Matrix implements Cloneable, java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Array for internal storage of elements.@serial internal array storage.*/private double[][] A;/** Row and column dimensions.@serial row dimension.@serial column dimension.*/private int m, n;/* ------------------------Constructors* ------------------------ *//** Construct an m-by-n matrix of zeros. @param m Number of rows.@param n Number of colums.*/public Matrix (int m, int n) {this.m = m;this.n = n;A = new double[m][n];}/** Construct an m-by-n constant matrix.@param m Number of rows.@param n Number of colums.@param s Fill the matrix with this scalar value.*/public Matrix (int m, int n, double s) {this.m = m;this.n = n;A = new double[m][n];for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = s;}}}/** Construct a matrix from a 2-D array.@param A Two-dimensional array of doubles.@exception IllegalArgumentException All rows must have the same length@see #constructWithCopy*/public Matrix (double[][] A) {m = A.length;n = A[0].length;for (int i = 0; i < m; i++) {if (A[i].length != n) {throw new IllegalArgumentException("All rows must have the same length.");}}this.A = A;}/** Construct a matrix quickly without checking arguments.@param A Two-dimensional array of doubles.@param m Number of rows.@param n Number of colums.*/public Matrix (double[][] A, int m, int n) {this.A = A;this.m = m;this.n = n;}/** Construct a matrix from a one-dimensional packed array@param vals One-dimensional array of doubles, packed by columns (ala Fortran).@param m Number of rows.@exception IllegalArgumentException Array length must be a multiple of m.*/public Matrix (double vals[], int m) {this.m = m;n = (m != 0 ? vals.length/m : 0);if (m*n != vals.length) {throw new IllegalArgumentException("Array length must be a multiple of m.");}A = new double[m][n];for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = vals[i+j*m];}}}/* ------------------------Public Methods* ------------------------ *//** Construct a matrix from a copy of a 2-D array.@param A Two-dimensional array of doubles.@exception IllegalArgumentException All rows must have the same length*/public static Matrix constructWithCopy(double[][] A) {int m = A.length;int n = A[0].length;Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {if (A[i].length != n) {throw new IllegalArgumentException("All rows must have the same length.");}for (int j = 0; j < n; j++) {C[i][j] = A[i][j];}}return X;}/** Make a deep copy of a matrix*/public Matrix copy () {Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j];}}return X;}/** Clone the Matrix object.*/public Object clone () {return this.copy();}/** Access the internal two-dimensional array.@return Pointer to the two-dimensional array of matrix elements.*/public double[][] getArray () {return A;}/** Copy the internal two-dimensional array.@return Two-dimensional array copy of matrix elements.*/public double[][] getArrayCopy () {double[][] C = new double[m][n];for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j];}}return C;}/** Make a one-dimensional column packed copy of the internal array.@return Matrix elements packed in a one-dimensional array by columns.*/public double[] getColumnPackedCopy () {double[] vals = new double[m*n];for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {vals[i+j*m] = A[i][j];}}return vals;}/** Make a one-dimensional row packed copy of the internal array.@return Matrix elements packed in a one-dimensional array by rows.*/public double[] getRowPackedCopy () {double[] vals = new double[m*n];for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {vals[i*n+j] = A[i][j];}}return vals;}/** Get row dimension.@return m, the number of rows.*/public int getRowDimension () {return m;}/** Get column dimension.@return n, the number of columns.*/public int getColumnDimension () {return n;}/** Get a single element.@param i Row index.@param j Column index.@return A(i,j)@exception ArrayIndexOutOfBoundsException*/public double get (int i, int j) {return A[i][j];}/** Get a submatrix.@param i0 Initial row index@param i1 Final row index@param j0 Initial column index@param j1 Final column index@return A(i0:i1,j0:j1)@exception ArrayIndexOutOfBoundsException Submatrix indices*/public Matrix getMatrix (int i0, int i1, int j0, int j1) {Matrix X = new Matrix(i1-i0+1,j1-j0+1);double[][] B = X.getArray();try {for (int i = i0; i <= i1; i++) {for (int j = j0; j <= j1; j++) {B[i-i0][j-j0] = A[i][j];}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}return X;}/** Get a submatrix.@param r Array of row indices.@param c Array of column indices.@return A(r(:),c(:))@exception ArrayIndexOutOfBoundsException Submatrix indices*/public Matrix getMatrix (int[] r, int[] c) {Matrix X = new Matrix(r.length,c.length);double[][] B = X.getArray();try {for (int i = 0; i < r.length; i++) {for (int j = 0; j < c.length; j++) {B[i][j] = A[r[i]][c[j]];}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}return X;}/** Get a submatrix.@param i0 Initial row index@param i1 Final row index@param c Array of column indices.@return A(i0:i1,c(:))@exception ArrayIndexOutOfBoundsException Submatrix indices*/public Matrix getMatrix (int i0, int i1, int[] c) {Matrix X = new Matrix(i1-i0+1,c.length);double[][] B = X.getArray();try {for (int i = i0; i <= i1; i++) {for (int j = 0; j < c.length; j++) {B[i-i0][j] = A[i][c[j]];}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}return X;}/** Get a submatrix.@param r Array of row indices.@param j0 Initial column index@param j1 Final column index@return A(r(:),j0:j1)@exception ArrayIndexOutOfBoundsException Submatrix indices*/public Matrix getMatrix (int[] r, int j0, int j1) {Matrix X = new Matrix(r.length,j1-j0+1);double[][] B = X.getArray();try {for (int i = 0; i < r.length; i++) {for (int j = j0; j <= j1; j++) {B[i][j-j0] = A[r[i]][j];}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}return X;}/** Set a single element.@param i Row index.@param j Column index.@param s A(i,j).@exception ArrayIndexOutOfBoundsException*/public void set (int i, int j, double s) {A[i][j] = s;}/** Set a submatrix.@param i0 Initial row index@param i1 Final row index@param j0 Initial column index@param j1 Final column index@param X A(i0:i1,j0:j1)@exception ArrayIndexOutOfBoundsException Submatrix indices*/public void setMatrix (int i0, int i1, int j0, int j1, Matrix X) {try {for (int i = i0; i <= i1; i++) {for (int j = j0; j <= j1; j++) {A[i][j] = X.get(i-i0,j-j0);}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}}/** Set a submatrix.@param r Array of row indices.@param c Array of column indices.@param X A(r(:),c(:))@exception ArrayIndexOutOfBoundsException Submatrix indices*/public void setMatrix (int[] r, int[] c, Matrix X) {try {for (int i = 0; i < r.length; i++) {for (int j = 0; j < c.length; j++) {A[r[i]][c[j]] = X.get(i,j);}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}}/** Set a submatrix.@param r Array of row indices.@param j0 Initial column index@param j1 Final column index@param X A(r(:),j0:j1)@exception ArrayIndexOutOfBoundsException Submatrix indices*/public void setMatrix (int[] r, int j0, int j1, Matrix X) {try {for (int i = 0; i < r.length; i++) {for (int j = j0; j <= j1; j++) {A[r[i]][j] = X.get(i,j-j0);}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}}/** Set a submatrix.@param i0 Initial row index@param i1 Final row index@param c Array of column indices.@param X A(i0:i1,c(:))@exception ArrayIndexOutOfBoundsException Submatrix indices*/public void setMatrix (int i0, int i1, int[] c, Matrix X) {try {for (int i = i0; i <= i1; i++) {for (int j = 0; j < c.length; j++) {A[i][c[j]] = X.get(i-i0,j);}}} catch(ArrayIndexOutOfBoundsException e) {throw new ArrayIndexOutOfBoundsException("Submatrix indices");}}/** Matrix transpose.@return A'*/public Matrix transpose () {Matrix X = new Matrix(n,m);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[j][i] = A[i][j];}}return X;}/** One norm@return maximum column sum.*/public double norm1 () {double f = 0;for (int j = 0; j < n; j++) {double s = 0;for (int i = 0; i < m; i++) {s += Math.abs(A[i][j]);}f = Math.max(f,s);}return f;}/** Two norm@return maximum singular value.*/public double norm2 () {return (new SingularValueDecomposition(this).norm2());}/** Infinity norm@return maximum row sum.*/public double normInf () {double f = 0;for (int i = 0; i < m; i++) {double s = 0;for (int j = 0; j < n; j++) {s += Math.abs(A[i][j]);}f = Math.max(f,s);}return f;}/** Frobenius norm@return sqrt of sum of squares of all elements.*/public double normF () {double f = 0;for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {f = Maths.hypot(f,A[i][j]);}}return f;}/** Unary minus@return -A*/public Matrix uminus () {Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = -A[i][j];}}return X;}/** C = A + B@param B another matrix@return A + B*/public Matrix plus (Matrix B) {checkMatrixDimensions(B);Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j] + B.A[i][j];}}return X;}/** A = A + B@param B another matrix@return A + B*/public Matrix plusEquals (Matrix B) {checkMatrixDimensions(B);for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = A[i][j] + B.A[i][j];}}return this;}/** C = A - B@param B another matrix@return A - B*/public Matrix minus (Matrix B) {checkMatrixDimensions(B);Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j] - B.A[i][j];}}return X;}/** A = A - B@param B another matrix@return A - B*/public Matrix minusEquals (Matrix B) {checkMatrixDimensions(B);for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = A[i][j] - B.A[i][j];}}return this;}/** Element-by-element multiplication, C = A.*B@param B another matrix@return A.*B*/public Matrix arrayTimes (Matrix B) {checkMatrixDimensions(B);Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j] * B.A[i][j];}}return X;}/** Element-by-element multiplication in place, A = A.*B@param B another matrix@return A.*B*/public Matrix arrayTimesEquals (Matrix B) {checkMatrixDimensions(B);for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = A[i][j] * B.A[i][j];}}return this;}/** Element-by-element right division, C = A./B@param B another matrix@return A./B*/public Matrix arrayRightDivide (Matrix B) {checkMatrixDimensions(B);Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = A[i][j] / B.A[i][j];}}return X;}/** Element-by-element right division in place, A = A./B@param B another matrix@return A./B*/public Matrix arrayRightDivideEquals (Matrix B) {checkMatrixDimensions(B);for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = A[i][j] / B.A[i][j];}}return this;}/** Element-by-element left division, C = A.\B@param B another matrix@return A.\B*/public Matrix arrayLeftDivide (Matrix B) {checkMatrixDimensions(B);Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = B.A[i][j] / A[i][j];}}return X;}/** Element-by-element left division in place, A = A.\B@param B another matrix@return A.\B*/public Matrix arrayLeftDivideEquals (Matrix B) {checkMatrixDimensions(B);for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = B.A[i][j] / A[i][j];}}return this;}/** Multiply a matrix by a scalar, C = s*A@param s scalar@return s*A*/public Matrix times (double s) {Matrix X = new Matrix(m,n);double[][] C = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {C[i][j] = s*A[i][j];}}return X;}/** Multiply a matrix by a scalar in place, A = s*A@param s scalar@return replace A by s*A*/public Matrix timesEquals (double s) {for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {A[i][j] = s*A[i][j];}}return this;}/** Linear algebraic matrix multiplication, A * B@param B another matrix@return Matrix product, A * B@exception IllegalArgumentException Matrix inner dimensions must agree.*/public Matrix times (Matrix B) {if (B.m != n) {throw new IllegalArgumentException("Matrix inner dimensions must agree.");}Matrix X = new Matrix(m,B.n);double[][] C = X.getArray();double[] Bcolj = new double[n];for (int j = 0; j < B.n; j++) {for (int k = 0; k < n; k++) {Bcolj[k] = B.A[k][j];}for (int i = 0; i < m; i++) {double[] Arowi = A[i];double s = 0;for (int k = 0; k < n; k++) {s += Arowi[k]*Bcolj[k];}C[i][j] = s;}}return X;}/** LU Decomposition@return LUDecomposition@see LUDecomposition*/public LUDecomposition lu () {return new LUDecomposition(this);}/** QR Decomposition@return QRDecomposition@see QRDecomposition*/public QRDecomposition qr () {return new QRDecomposition(this);}/** Cholesky Decomposition@return CholeskyDecomposition@see CholeskyDecomposition*/public CholeskyDecomposition chol () {return new CholeskyDecomposition(this);}/** Singular Value Decomposition@return SingularValueDecomposition@see SingularValueDecomposition*/public SingularValueDecomposition svd () {return new SingularValueDecomposition(this);}/** Eigenvalue Decomposition@return EigenvalueDecomposition@see EigenvalueDecomposition*/public EigenvalueDecomposition eig () {return new EigenvalueDecomposition(this);}/** Solve A*X = B@param B right hand side@return solution if A is square, least squares solution otherwise*/public Matrix solve (Matrix B) {return (m == n ? (new LUDecomposition(this)).solve(B) :(new QRDecomposition(this)).solve(B));}/** Solve X*A = B, which is also A'*X' = B'@param B right hand side@return solution if A is square, least squares solution otherwise.*/public Matrix solveTranspose (Matrix B) {return transpose().solve(B.transpose());}/** Matrix inverse or pseudoinverse@return inverse(A) if A is square, pseudoinverse otherwise.*/public Matrix inverse () {return solve(identity(m,m));}/** Matrix determinant@return determinant*/public double det () {return new LUDecomposition(this).det();}/** Matrix rank@return effective numerical rank, obtained from SVD.*/public int rank () {return new SingularValueDecomposition(this).rank();}/** Matrix condition (2 norm)@return ratio of largest to smallest singular value.*/public double cond () {return new SingularValueDecomposition(this).cond();}/** Matrix trace.@return sum of the diagonal elements.*/public double trace () {double t = 0;for (int i = 0; i < Math.min(m,n); i++) {t += A[i][i];}return t;}/** Generate matrix with random elements@param m Number of rows.@param n Number of colums.@return An m-by-n matrix with uniformly distributed random elements.*/public static Matrix random (int m, int n) {Matrix A = new Matrix(m,n);double[][] X = A.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {X[i][j] = Math.random();}}return A;}/** Generate identity matrix@param m Number of rows.@param n Number of colums.@return An m-by-n matrix with ones on the diagonal and zeros elsewhere.*/public static Matrix identity (int m, int n) {Matrix A = new Matrix(m,n);double[][] X = A.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {X[i][j] = (i == j ? 1.0 : 0.0);}}return A;}/** Print the matrix to stdout. Line the elements up in columns* with a Fortran-like 'Fw.d' style format.@param w Column width.@param d Number of digits after the decimal.*/public void print (int w, int d) {print(new PrintWriter(System.out,true),w,d); }/** Print the matrix to the output stream. Line the elements up in* columns with a Fortran-like 'Fw.d' style format.@param output Output stream.@param w Column width.@param d Number of digits after the decimal.*/public void print (PrintWriter output, int w, int d) {DecimalFormat format = new DecimalFormat();format.setDecimalFormatSymbols(new DecimalFormatSymbols(Locale.US));format.setMinimumIntegerDigits(1);format.setMaximumFractionDigits(d);format.setMinimumFractionDigits(d);format.setGroupingUsed(false);print(output,format,w+2);}/** Print the matrix to stdout. Line the elements up in columns.* Use the format object, and right justify within columns of width* characters.* Note that is the matrix is to be read back in, you probably will want* to use a NumberFormat that is set to US Locale.@param format A Formatting object for individual elements.@param width Field width for each column.@see java.text.DecimalFormat#setDecimalFormatSymbols*/public void print (NumberFormat format, int width) {print(new PrintWriter(System.out,true),format,width); }// DecimalFormat is a little disappointing coming from Fortran or C's printf.// Since it doesn't pad on the left, the elements will come out different// widths. Consequently, we'll pass the desired column width in as an// argument and do the extra padding ourselves./** Print the matrix to the output stream. Line the elements up in columns.* Use the format object, and right justify within columns of width* characters.* Note that is the matrix is to be read back in, you probably will want* to use a NumberFormat that is set to US Locale.@param output the output stream.@param format A formatting object to format the matrix elements @param width Column width.@see java.text.DecimalFormat#setDecimalFormatSymbols*/public void print (PrintWriter output, NumberFormat format, int width) {output.println(); // start on new line.for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {String s = format.format(A[i][j]); // format the numberint padding = Math.max(1,width-s.length()); // At _least_ 1 spacefor (int k = 0; k < padding; k++)output.print(' ');output.print(s);}output.println();}output.println(); // end with blank line.}/** Read a matrix from a stream. The format is the same the print method,* so printed matrices can be read back in (provided they were printed using* US Locale). Elements are separated by* whitespace, all the elements for each row appear on a single line,* the last row is followed by a blank line.@param input the input stream.*/public static Matrix read (BufferedReader input) throws java.io.IOException {StreamTokenizer tokenizer= new StreamTokenizer(input);// Although StreamTokenizer will parse numbers, it doesn't recognize// scientific notation (E or D); however, Double.valueOf does.// The strategy here is to disable StreamTokenizer's number parsing.// We'll only get whitespace delimited words, EOL's and EOF's.// These words should all be numbers, for Double.valueOf to parse.tokenizer.resetSyntax();tokenizer.wordChars(0,255);tokenizer.whitespaceChars(0, ' ');tokenizer.eolIsSignificant(true);java.util.Vector<Double> vD = new java.util.Vector<Double>();// Ignore initial empty lineswhile (tokenizer.nextToken() == StreamTokenizer.TT_EOL);if (tokenizer.ttype == StreamTokenizer.TT_EOF)throw new java.io.IOException("Unexpected EOF on matrix read.");do {vD.addElement(Double.valueOf(tokenizer.sval)); // Read & store 1st row.} while (tokenizer.nextToken() == StreamTokenizer.TT_WORD);int n = vD.size(); // Now we've got the number of columns!double row[] = new double[n];for (int j=0; j<n; j++) // extract the elements of the 1st row.row[j]=vD.elementAt(j).doubleValue();java.util.Vector<double[]> v = new java.util.Vector<double[]>();v.addElement(row); // Start storing rows instead of columns.while (tokenizer.nextToken() == StreamTokenizer.TT_WORD) {// While non-empty linesv.addElement(row = new double[n]);int j = 0;do {if (j >= n) throw new java.io.IOException("Row " + v.size() + " is too long.");row[j++] = Double.valueOf(tokenizer.sval).doubleValue();} while (tokenizer.nextToken() == StreamTokenizer.TT_WORD);if (j < n) throw new java.io.IOException("Row " + v.size() + " is too short.");}int m = v.size(); // Now we've got the number of rows.double[][] A = new double[m][];v.copyInto(A); // copy the rows out of the vectorreturn new Matrix(A);}/* ------------------------Private Methods* ------------------------ *//** Check if size(A) == size(B) **/private void checkMatrixDimensions (Matrix B) {if (B.m != m || B.n != n) {throw new IllegalArgumentException("Matrix dimensions must agree.");}}private static final long serialVersionUID = 1; }
package Jama.util;public class Maths {/** sqrt(a^2 + b^2) without under/overflow. **/public static double hypot(double a, double b) {double r;if (Math.abs(a) > Math.abs(b)) {r = b/a;r = Math.abs(a)*Math.sqrt(1+r*r);} else if (b != 0) {r = a/b;r = Math.abs(b)*Math.sqrt(1+r*r);} else {r = 0.0;}return r;} }

package Jama; import Jama.util.*;/** Singular Value Decomposition.<P>For an m-by-n matrix A with m >= n, the singular value decomposition isan m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, andan n-by-n orthogonal matrix V so that A = U*S*V'.<P>The singular values, sigma[k] = S[k][k], are ordered so thatsigma[0] >= sigma[1] >= ... >= sigma[n-1].<P>The singular value decompostion always exists, so the constructor willnever fail. The matrix condition number and the effective numericalrank can be computed from this decomposition.*/public class SingularValueDecomposition implements java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Arrays for internal storage of U and V.@serial internal storage of U.@serial internal storage of V.*/private double[][] U, V;/** Array for internal storage of singular values.@serial internal storage of singular values.*/private double[] s;/** Row and column dimensions.@serial row dimension.@serial column dimension.*/private int m, n;/* ------------------------Constructor* ------------------------ *//** Construct the singular value decompositionStructure to access U, S and V.@param Arg Rectangular matrix*/public SingularValueDecomposition (Matrix Arg) {// Derived from LINPACK code.// Initialize.double[][] A = Arg.getArrayCopy();m = Arg.getRowDimension();n = Arg.getColumnDimension();/* Apparently the failing cases are only a proper subset of (m<n), so let's not throw error. Correct fix to come later?if (m<n) {throw new IllegalArgumentException("Jama SVD only works for m >= n"); }*/int nu = Math.min(m,n);s = new double [Math.min(m+1,n)];U = new double [m][nu];V = new double [n][n];double[] e = new double [n];double[] work = new double [m];boolean wantu = true;boolean wantv = true;// Reduce A to bidiagonal form, storing the diagonal elements// in s and the super-diagonal elements in e.int nct = Math.min(m-1,n);int nrt = Math.max(0,Math.min(n-2,m));for (int k = 0; k < Math.max(nct,nrt); k++) {if (k < nct) {// Compute the transformation for the k-th column and// place the k-th diagonal in s[k].// Compute 2-norm of k-th column without under/overflow.s[k] = 0;for (int i = k; i < m; i++) {s[k] = Maths.hypot(s[k],A[i][k]);}if (s[k] != 0.0) {if (A[k][k] < 0.0) {s[k] = -s[k];}for (int i = k; i < m; i++) {A[i][k] /= s[k];}A[k][k] += 1.0;}s[k] = -s[k];}for (int j = k+1; j < n; j++) {if ((k < nct) & (s[k] != 0.0)) {// Apply the transformation.double t = 0;for (int i = k; i < m; i++) {t += A[i][k]*A[i][j];}t = -t/A[k][k];for (int i = k; i < m; i++) {A[i][j] += t*A[i][k];}}// Place the k-th row of A into e for the// subsequent calculation of the row transformation.e[j] = A[k][j];}if (wantu & (k < nct)) {// Place the transformation in U for subsequent back// multiplication.for (int i = k; i < m; i++) {U[i][k] = A[i][k];}}if (k < nrt) {// Compute the k-th row transformation and place the// k-th super-diagonal in e[k].// Compute 2-norm without under/overflow.e[k] = 0;for (int i = k+1; i < n; i++) {e[k] = Maths.hypot(e[k],e[i]);}if (e[k] != 0.0) {if (e[k+1] < 0.0) {e[k] = -e[k];}for (int i = k+1; i < n; i++) {e[i] /= e[k];}e[k+1] += 1.0;}e[k] = -e[k];if ((k+1 < m) & (e[k] != 0.0)) {// Apply the transformation.for (int i = k+1; i < m; i++) {work[i] = 0.0;}for (int j = k+1; j < n; j++) {for (int i = k+1; i < m; i++) {work[i] += e[j]*A[i][j];}}for (int j = k+1; j < n; j++) {double t = -e[j]/e[k+1];for (int i = k+1; i < m; i++) {A[i][j] += t*work[i];}}}if (wantv) {// Place the transformation in V for subsequent// back multiplication.for (int i = k+1; i < n; i++) {V[i][k] = e[i];}}}}// Set up the final bidiagonal matrix or order p.int p = Math.min(n,m+1);if (nct < n) {s[nct] = A[nct][nct];}if (m < p) {s[p-1] = 0.0;}if (nrt+1 < p) {e[nrt] = A[nrt][p-1];}e[p-1] = 0.0;// If required, generate U.if (wantu) {for (int j = nct; j < nu; j++) {for (int i = 0; i < m; i++) {U[i][j] = 0.0;}U[j][j] = 1.0;}for (int k = nct-1; k >= 0; k--) {if (s[k] != 0.0) {for (int j = k+1; j < nu; j++) {double t = 0;for (int i = k; i < m; i++) {t += U[i][k]*U[i][j];}t = -t/U[k][k];for (int i = k; i < m; i++) {U[i][j] += t*U[i][k];}}for (int i = k; i < m; i++ ) {U[i][k] = -U[i][k];}U[k][k] = 1.0 + U[k][k];for (int i = 0; i < k-1; i++) {U[i][k] = 0.0;}} else {for (int i = 0; i < m; i++) {U[i][k] = 0.0;}U[k][k] = 1.0;}}}// If required, generate V.if (wantv) {for (int k = n-1; k >= 0; k--) {if ((k < nrt) & (e[k] != 0.0)) {for (int j = k+1; j < nu; j++) {double t = 0;for (int i = k+1; i < n; i++) {t += V[i][k]*V[i][j];}t = -t/V[k+1][k];for (int i = k+1; i < n; i++) {V[i][j] += t*V[i][k];}}}for (int i = 0; i < n; i++) {V[i][k] = 0.0;}V[k][k] = 1.0;}}// Main iteration loop for the singular values.int pp = p-1;int iter = 0;double eps = Math.pow(2.0,-52.0);double tiny = Math.pow(2.0,-966.0);while (p > 0) {int k,kase;// Here is where a test for too many iterations would go.// This section of the program inspects for// negligible elements in the s and e arrays. On// completion the variables kase and k are set as follows.// kase = 1 if s(p) and e[k-1] are negligible and k<p// kase = 2 if s(k) is negligible and k<p// kase = 3 if e[k-1] is negligible, k<p, and// s(k), ..., s(p) are not negligible (qr step).// kase = 4 if e(p-1) is negligible (convergence).for (k = p-2; k >= -1; k--) {if (k == -1) {break;}if (Math.abs(e[k]) <=tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))) {e[k] = 0.0;break;}}if (k == p-2) {kase = 4;} else {int ks;for (ks = p-1; ks >= k; ks--) {if (ks == k) {break;}double t = (ks != p ? Math.abs(e[ks]) : 0.) + (ks != k+1 ? Math.abs(e[ks-1]) : 0.);if (Math.abs(s[ks]) <= tiny + eps*t) {s[ks] = 0.0;break;}}if (ks == k) {kase = 3;} else if (ks == p-1) {kase = 1;} else {kase = 2;k = ks;}}k++;// Perform the task indicated by kase.switch (kase) {// Deflate negligible s(p).case 1: {double f = e[p-2];e[p-2] = 0.0;for (int j = p-2; j >= k; j--) {double t = Maths.hypot(s[j],f);double cs = s[j]/t;double sn = f/t;s[j] = t;if (j != k) {f = -sn*e[j-1];e[j-1] = cs*e[j-1];}if (wantv) {for (int i = 0; i < n; i++) {t = cs*V[i][j] + sn*V[i][p-1];V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];V[i][j] = t;}}}}break;// Split at negligible s(k).case 2: {double f = e[k-1];e[k-1] = 0.0;for (int j = k; j < p; j++) {double t = Maths.hypot(s[j],f);double cs = s[j]/t;double sn = f/t;s[j] = t;f = -sn*e[j];e[j] = cs*e[j];if (wantu) {for (int i = 0; i < m; i++) {t = cs*U[i][j] + sn*U[i][k-1];U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];U[i][j] = t;}}}}break;// Perform one qr step.case 3: {// Calculate the shift.double scale = Math.max(Math.max(Math.max(Math.max(Math.abs(s[p-1]),Math.abs(s[p-2])),Math.abs(e[p-2])), Math.abs(s[k])),Math.abs(e[k]));double sp = s[p-1]/scale;double spm1 = s[p-2]/scale;double epm1 = e[p-2]/scale;double sk = s[k]/scale;double ek = e[k]/scale;double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;double c = (sp*epm1)*(sp*epm1);double shift = 0.0;if ((b != 0.0) | (c != 0.0)) {shift = Math.sqrt(b*b + c);if (b < 0.0) {shift = -shift;}shift = c/(b + shift);}double f = (sk + sp)*(sk - sp) + shift;double g = sk*ek;// Chase zeros.for (int j = k; j < p-1; j++) {double t = Maths.hypot(f,g);double cs = f/t;double sn = g/t;if (j != k) {e[j-1] = t;}f = cs*s[j] + sn*e[j];e[j] = cs*e[j] - sn*s[j];g = sn*s[j+1];s[j+1] = cs*s[j+1];if (wantv) {for (int i = 0; i < n; i++) {t = cs*V[i][j] + sn*V[i][j+1];V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];V[i][j] = t;}}t = Maths.hypot(f,g);cs = f/t;sn = g/t;s[j] = t;f = cs*e[j] + sn*s[j+1];s[j+1] = -sn*e[j] + cs*s[j+1];g = sn*e[j+1];e[j+1] = cs*e[j+1];if (wantu && (j < m-1)) {for (int i = 0; i < m; i++) {t = cs*U[i][j] + sn*U[i][j+1];U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];U[i][j] = t;}}}e[p-2] = f;iter = iter + 1;}break;// Convergence.case 4: {// Make the singular values positive.if (s[k] <= 0.0) {s[k] = (s[k] < 0.0 ? -s[k] : 0.0);if (wantv) {for (int i = 0; i <= pp; i++) {V[i][k] = -V[i][k];}}}// Order the singular values.while (k < pp) {if (s[k] >= s[k+1]) {break;}double t = s[k];s[k] = s[k+1];s[k+1] = t;if (wantv && (k < n-1)) {for (int i = 0; i < n; i++) {t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;}}if (wantu && (k < m-1)) {for (int i = 0; i < m; i++) {t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;}}k++;}iter = 0;p--;}break;}}}/* ------------------------Public Methods* ------------------------ *//** Return the left singular vectors@return U*/public Matrix getU () {return new Matrix(U,m,Math.min(m+1,n));}/** Return the right singular vectors@return V*/public Matrix getV () {return new Matrix(V,n,n);}/** Return the one-dimensional array of singular values@return diagonal of S.*/public double[] getSingularValues () {return s;}/** Return the diagonal matrix of singular values@return S*/public Matrix getS () {Matrix X = new Matrix(n,n);double[][] S = X.getArray();for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {S[i][j] = 0.0;}S[i][i] = this.s[i];}return X;}/** Two norm@return max(S)*/public double norm2 () {return s[0];}/** Two norm condition number@return max(S)/min(S)*/public double cond () {return s[0]/s[Math.min(m,n)-1];}/** Effective numerical matrix rank@return Number of nonnegligible singular values.*/public int rank () {double eps = Math.pow(2.0,-52.0);double tol = Math.max(m,n)*s[0]*eps;int r = 0;for (int i = 0; i < s.length; i++) {if (s[i] > tol) {r++;}}return r;}private static final long serialVersionUID = 1; }
package Jama; import Jama.util.*;/** QR Decomposition. <P>For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-northogonal matrix Q and an n-by-n upper triangular matrix R so thatA = Q*R. <P>The QR decompostion always exists, even if the matrix does not havefull rank, so the constructor will never fail. The primary use of theQR decomposition is in the least squares solution of nonsquare systemsof simultaneous linear equations. This will fail if isFullRank()returns false. */public class QRDecomposition implements java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Array for internal storage of decomposition.@serial internal array storage.*/private double[][] QR;/** Row and column dimensions.@serial column dimension.@serial row dimension.*/private int m, n;/** Array for internal storage of diagonal of R.@serial diagonal of R.*/private double[] Rdiag;/* ------------------------Constructor* ------------------------ *//** QR Decomposition, computed by Householder reflections.Structure to access R and the Householder vectors and compute Q.@param A Rectangular matrix*/public QRDecomposition (Matrix A) {// Initialize.QR = A.getArrayCopy();m = A.getRowDimension();n = A.getColumnDimension();Rdiag = new double[n];// Main loop.for (int k = 0; k < n; k++) {// Compute 2-norm of k-th column without under/overflow.double nrm = 0;for (int i = k; i < m; i++) {nrm = Maths.hypot(nrm,QR[i][k]);}if (nrm != 0.0) {// Form k-th Householder vector.if (QR[k][k] < 0) {nrm = -nrm;}for (int i = k; i < m; i++) {QR[i][k] /= nrm;}QR[k][k] += 1.0;// Apply transformation to remaining columns.for (int j = k+1; j < n; j++) {double s = 0.0; for (int i = k; i < m; i++) {s += QR[i][k]*QR[i][j];}s = -s/QR[k][k];for (int i = k; i < m; i++) {QR[i][j] += s*QR[i][k];}}}Rdiag[k] = -nrm;}}/* ------------------------Public Methods* ------------------------ *//** Is the matrix full rank?@return true if R, and hence A, has full rank.*/public boolean isFullRank () {for (int j = 0; j < n; j++) {if (Rdiag[j] == 0)return false;}return true;}/** Return the Householder vectors@return Lower trapezoidal matrix whose columns define the reflections*/public Matrix getH () {Matrix X = new Matrix(m,n);double[][] H = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {if (i >= j) {H[i][j] = QR[i][j];} else {H[i][j] = 0.0;}}}return X;}/** Return the upper triangular factor@return R*/public Matrix getR () {Matrix X = new Matrix(n,n);double[][] R = X.getArray();for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {if (i < j) {R[i][j] = QR[i][j];} else if (i == j) {R[i][j] = Rdiag[i];} else {R[i][j] = 0.0;}}}return X;}/** Generate and return the (economy-sized) orthogonal factor@return Q*/public Matrix getQ () {Matrix X = new Matrix(m,n);double[][] Q = X.getArray();for (int k = n-1; k >= 0; k--) {for (int i = 0; i < m; i++) {Q[i][k] = 0.0;}Q[k][k] = 1.0;for (int j = k; j < n; j++) {if (QR[k][k] != 0) {double s = 0.0;for (int i = k; i < m; i++) {s += QR[i][k]*Q[i][j];}s = -s/QR[k][k];for (int i = k; i < m; i++) {Q[i][j] += s*QR[i][k];}}}}return X;}/** Least squares solution of A*X = B@param B A Matrix with as many rows as A and any number of columns.@return X that minimizes the two norm of Q*R*X-B.@exception IllegalArgumentException Matrix row dimensions must agree.@exception RuntimeException Matrix is rank deficient.*/public Matrix solve (Matrix B) {if (B.getRowDimension() != m) {throw new IllegalArgumentException("Matrix row dimensions must agree.");}if (!this.isFullRank()) {throw new RuntimeException("Matrix is rank deficient.");}// Copy right hand sideint nx = B.getColumnDimension();double[][] X = B.getArrayCopy();// Compute Y = transpose(Q)*Bfor (int k = 0; k < n; k++) {for (int j = 0; j < nx; j++) {double s = 0.0; for (int i = k; i < m; i++) {s += QR[i][k]*X[i][j];}s = -s/QR[k][k];for (int i = k; i < m; i++) {X[i][j] += s*QR[i][k];}}}// Solve R*X = Y;for (int k = n-1; k >= 0; k--) {for (int j = 0; j < nx; j++) {X[k][j] /= Rdiag[k];}for (int i = 0; i < k; i++) {for (int j = 0; j < nx; j++) {X[i][j] -= X[k][j]*QR[i][k];}}}return (new Matrix(X,n,nx).getMatrix(0,n-1,0,nx-1));}private static final long serialVersionUID = 1; }
package Jama;/** LU Decomposition.<P>For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-nunit lower triangular matrix L, an n-by-n upper triangular matrix U,and a permutation vector piv of length m so that A(piv,:) = L*U.If m < n, then L is m-by-m and U is m-by-n.<P>The LU decompostion with pivoting always exists, even if the matrix issingular, so the constructor will never fail. The primary use of theLU decomposition is in the solution of square systems of simultaneouslinear equations. This will fail if isNonsingular() returns false.*/public class LUDecomposition implements java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Array for internal storage of decomposition.@serial internal array storage.*/private double[][] LU;/** Row and column dimensions, and pivot sign.@serial column dimension.@serial row dimension.@serial pivot sign.*/private int m, n, pivsign; /** Internal storage of pivot vector.@serial pivot vector.*/private int[] piv;/* ------------------------Constructor* ------------------------ *//** LU DecompositionStructure to access L, U and piv.@param A Rectangular matrix*/public LUDecomposition (Matrix A) {// Use a "left-looking", dot-product, Crout/Doolittle algorithm.LU = A.getArrayCopy();m = A.getRowDimension();n = A.getColumnDimension();piv = new int[m];for (int i = 0; i < m; i++) {piv[i] = i;}pivsign = 1;double[] LUrowi;double[] LUcolj = new double[m];// Outer loop.for (int j = 0; j < n; j++) {// Make a copy of the j-th column to localize references.for (int i = 0; i < m; i++) {LUcolj[i] = LU[i][j];}// Apply previous transformations.for (int i = 0; i < m; i++) {LUrowi = LU[i];// Most of the time is spent in the following dot product.int kmax = Math.min(i,j);double s = 0.0;for (int k = 0; k < kmax; k++) {s += LUrowi[k]*LUcolj[k];}LUrowi[j] = LUcolj[i] -= s;}// Find pivot and exchange if necessary.int p = j;for (int i = j+1; i < m; i++) {if (Math.abs(LUcolj[i]) > Math.abs(LUcolj[p])) {p = i;}}if (p != j) {for (int k = 0; k < n; k++) {double t = LU[p][k]; LU[p][k] = LU[j][k]; LU[j][k] = t;}int k = piv[p]; piv[p] = piv[j]; piv[j] = k;pivsign = -pivsign;}// Compute multipliers.if (j < m & LU[j][j] != 0.0) {for (int i = j+1; i < m; i++) {LU[i][j] /= LU[j][j];}}}}/* ------------------------Temporary, experimental code.------------------------ *\\** LU Decomposition, computed by Gaussian elimination.<P>This constructor computes L and U with the "daxpy"-based eliminationalgorithm used in LINPACK and MATLAB. In Java, we suspect the dot-product,Crout algorithm will be faster. We have temporarily included thisconstructor until timing experiments confirm this suspicion.<P>@param A Rectangular matrix@param linpackflag Use Gaussian elimination. Actual value ignored.@return Structure to access L, U and piv.*\public LUDecomposition (Matrix A, int linpackflag) {// Initialize.LU = A.getArrayCopy();m = A.getRowDimension();n = A.getColumnDimension();piv = new int[m];for (int i = 0; i < m; i++) {piv[i] = i;}pivsign = 1;// Main loop.for (int k = 0; k < n; k++) {// Find pivot.int p = k;for (int i = k+1; i < m; i++) {if (Math.abs(LU[i][k]) > Math.abs(LU[p][k])) {p = i;}}// Exchange if necessary.if (p != k) {for (int j = 0; j < n; j++) {double t = LU[p][j]; LU[p][j] = LU[k][j]; LU[k][j] = t;}int t = piv[p]; piv[p] = piv[k]; piv[k] = t;pivsign = -pivsign;}// Compute multipliers and eliminate k-th column.if (LU[k][k] != 0.0) {for (int i = k+1; i < m; i++) {LU[i][k] /= LU[k][k];for (int j = k+1; j < n; j++) {LU[i][j] -= LU[i][k]*LU[k][j];}}}}}\* ------------------------End of temporary code.* ------------------------ *//* ------------------------Public Methods* ------------------------ *//** Is the matrix nonsingular?@return true if U, and hence A, is nonsingular.*/public boolean isNonsingular () {for (int j = 0; j < n; j++) {if (LU[j][j] == 0)return false;}return true;}/** Return lower triangular factor@return L*/public Matrix getL () {Matrix X = new Matrix(m,n);double[][] L = X.getArray();for (int i = 0; i < m; i++) {for (int j = 0; j < n; j++) {if (i > j) {L[i][j] = LU[i][j];} else if (i == j) {L[i][j] = 1.0;} else {L[i][j] = 0.0;}}}return X;}/** Return upper triangular factor@return U*/public Matrix getU () {Matrix X = new Matrix(n,n);double[][] U = X.getArray();for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {if (i <= j) {U[i][j] = LU[i][j];} else {U[i][j] = 0.0;}}}return X;}/** Return pivot permutation vector@return piv*/public int[] getPivot () {int[] p = new int[m];for (int i = 0; i < m; i++) {p[i] = piv[i];}return p;}/** Return pivot permutation vector as a one-dimensional double array@return (double) piv*/public double[] getDoublePivot () {double[] vals = new double[m];for (int i = 0; i < m; i++) {vals[i] = (double) piv[i];}return vals;}/** Determinant@return det(A)@exception IllegalArgumentException Matrix must be square*/public double det () {if (m != n) {throw new IllegalArgumentException("Matrix must be square.");}double d = (double) pivsign;for (int j = 0; j < n; j++) {d *= LU[j][j];}return d;}/** Solve A*X = B@param B A Matrix with as many rows as A and any number of columns.@return X so that L*U*X = B(piv,:)@exception IllegalArgumentException Matrix row dimensions must agree.@exception RuntimeException Matrix is singular.*/public Matrix solve (Matrix B) {if (B.getRowDimension() != m) {throw new IllegalArgumentException("Matrix row dimensions must agree.");}if (!this.isNonsingular()) {throw new RuntimeException("Matrix is singular.");}// Copy right hand side with pivotingint nx = B.getColumnDimension();Matrix Xmat = B.getMatrix(piv,0,nx-1);double[][] X = Xmat.getArray();// Solve L*Y = B(piv,:)for (int k = 0; k < n; k++) {for (int i = k+1; i < n; i++) {for (int j = 0; j < nx; j++) {X[i][j] -= X[k][j]*LU[i][k];}}}// Solve U*X = Y;for (int k = n-1; k >= 0; k--) {for (int j = 0; j < nx; j++) {X[k][j] /= LU[k][k];}for (int i = 0; i < k; i++) {for (int j = 0; j < nx; j++) {X[i][j] -= X[k][j]*LU[i][k];}}}return Xmat;}private static final long serialVersionUID = 1; }
package Jama;/** Cholesky Decomposition.<P>For a symmetric, positive definite matrix A, the Cholesky decompositionis an lower triangular matrix L so that A = L*L'.<P>If the matrix is not symmetric or positive definite, the constructorreturns a partial decomposition and sets an internal flag that maybe queried by the isSPD() method.*/public class CholeskyDecomposition implements java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Array for internal storage of decomposition.@serial internal array storage.*/private double[][] L;/** Row and column dimension (square matrix).@serial matrix dimension.*/private int n;/** Symmetric and positive definite flag.@serial is symmetric and positive definite flag.*/private boolean isspd;/* ------------------------Constructor* ------------------------ *//** Cholesky algorithm for symmetric and positive definite matrix.Structure to access L and isspd flag.@param Arg Square, symmetric matrix.*/public CholeskyDecomposition (Matrix Arg) {// Initialize.double[][] A = Arg.getArray();n = Arg.getRowDimension();L = new double[n][n];isspd = (Arg.getColumnDimension() == n);// Main loop.for (int j = 0; j < n; j++) {double[] Lrowj = L[j];double d = 0.0;for (int k = 0; k < j; k++) {double[] Lrowk = L[k];double s = 0.0;for (int i = 0; i < k; i++) {s += Lrowk[i]*Lrowj[i];}Lrowj[k] = s = (A[j][k] - s)/L[k][k];d = d + s*s;isspd = isspd & (A[k][j] == A[j][k]); }d = A[j][j] - d;isspd = isspd & (d > 0.0);L[j][j] = Math.sqrt(Math.max(d,0.0));for (int k = j+1; k < n; k++) {L[j][k] = 0.0;}}}/* ------------------------Temporary, experimental code.* ------------------------ *\\** Right Triangular Cholesky Decomposition.<P>For a symmetric, positive definite matrix A, the Right Choleskydecomposition is an upper triangular matrix R so that A = R'*R.This constructor computes R with the Fortran inspired column orientedalgorithm used in LINPACK and MATLAB. In Java, we suspect a row oriented,lower triangular decomposition is faster. We have temporarily includedthis constructor here until timing experiments confirm this suspicion.*\\** Array for internal storage of right triangular decomposition. **\private transient double[][] R;\** Cholesky algorithm for symmetric and positive definite matrix.@param A Square, symmetric matrix.@param rightflag Actual value ignored.@return Structure to access R and isspd flag.*\public CholeskyDecomposition (Matrix Arg, int rightflag) {// Initialize.double[][] A = Arg.getArray();n = Arg.getColumnDimension();R = new double[n][n];isspd = (Arg.getColumnDimension() == n);// Main loop.for (int j = 0; j < n; j++) {double d = 0.0;for (int k = 0; k < j; k++) {double s = A[k][j];for (int i = 0; i < k; i++) {s = s - R[i][k]*R[i][j];}R[k][j] = s = s/R[k][k];d = d + s*s;isspd = isspd & (A[k][j] == A[j][k]); }d = A[j][j] - d;isspd = isspd & (d > 0.0);R[j][j] = Math.sqrt(Math.max(d,0.0));for (int k = j+1; k < n; k++) {R[k][j] = 0.0;}}}\** Return upper triangular factor.@return R*\public Matrix getR () {return new Matrix(R,n,n);}\* ------------------------End of temporary code.* ------------------------ *//* ------------------------Public Methods* ------------------------ *//** Is the matrix symmetric and positive definite?@return true if A is symmetric and positive definite.*/public boolean isSPD () {return isspd;}/** Return triangular factor.@return L*/public Matrix getL () {return new Matrix(L,n,n);}/** Solve A*X = B@param B A Matrix with as many rows as A and any number of columns.@return X so that L*L'*X = B@exception IllegalArgumentException Matrix row dimensions must agree.@exception RuntimeException Matrix is not symmetric positive definite.*/public Matrix solve (Matrix B) {if (B.getRowDimension() != n) {throw new IllegalArgumentException("Matrix row dimensions must agree.");}if (!isspd) {throw new RuntimeException("Matrix is not symmetric positive definite.");}// Copy right hand side.double[][] X = B.getArrayCopy();int nx = B.getColumnDimension();// Solve L*Y = B;for (int k = 0; k < n; k++) {for (int j = 0; j < nx; j++) {for (int i = 0; i < k ; i++) {X[k][j] -= X[i][j]*L[k][i];}X[k][j] /= L[k][k];}}// Solve L'*X = Y;for (int k = n-1; k >= 0; k--) {for (int j = 0; j < nx; j++) {for (int i = k+1; i < n ; i++) {X[k][j] -= X[i][j]*L[i][k];}X[k][j] /= L[k][k];}}return new Matrix(X,n,nx);}private static final long serialVersionUID = 1;}
package Jama; import Jama.util.*;/** Eigenvalues and eigenvectors of a real matrix. <P>If A is symmetric, then A = V*D*V' where the eigenvalue matrix D isdiagonal and the eigenvector matrix V is orthogonal.I.e. A = V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the identity matrix. <P>If A is not symmetric, then the eigenvalue matrix D is block diagonalwith the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. Thecolumns of V represent the eigenvectors in the sense that A*V = V*D,i.e. A.times(V) equals V.times(D). The matrix V may be badlyconditioned, or even singular, so the validity of the equationA = V*D*inverse(V) depends upon V.cond(). **/public class EigenvalueDecomposition implements java.io.Serializable {/* ------------------------Class variables* ------------------------ *//** Row and column dimension (square matrix).@serial matrix dimension.*/private int n;/** Symmetry flag.@serial internal symmetry flag.*/private boolean issymmetric;/** Arrays for internal storage of eigenvalues.@serial internal storage of eigenvalues.*/private double[] d, e;/** Array for internal storage of eigenvectors.@serial internal storage of eigenvectors.*/private double[][] V;/** Array for internal storage of nonsymmetric Hessenberg form.@serial internal storage of nonsymmetric Hessenberg form.*/private double[][] H;/** Working storage for nonsymmetric algorithm.@serial working storage for nonsymmetric algorithm.*/private double[] ort;/* ------------------------Private Methods* ------------------------ */// Symmetric Householder reduction to tridiagonal form.private void tred2 () {// This is derived from the Algol procedures tred2 by// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding// Fortran subroutine in EISPACK.for (int j = 0; j < n; j++) {d[j] = V[n-1][j];}// Householder reduction to tridiagonal form.for (int i = n-1; i > 0; i--) {// Scale to avoid under/overflow.double scale = 0.0;double h = 0.0;for (int k = 0; k < i; k++) {scale = scale + Math.abs(d[k]);}if (scale == 0.0) {e[i] = d[i-1];for (int j = 0; j < i; j++) {d[j] = V[i-1][j];V[i][j] = 0.0;V[j][i] = 0.0;}} else {// Generate Householder vector.for (int k = 0; k < i; k++) {d[k] /= scale;h += d[k] * d[k];}double f = d[i-1];double g = Math.sqrt(h);if (f > 0) {g = -g;}e[i] = scale * g;h = h - f * g;d[i-1] = f - g;for (int j = 0; j < i; j++) {e[j] = 0.0;}// Apply similarity transformation to remaining columns.for (int j = 0; j < i; j++) {f = d[j];V[j][i] = f;g = e[j] + V[j][j] * f;for (int k = j+1; k <= i-1; k++) {g += V[k][j] * d[k];e[k] += V[k][j] * f;}e[j] = g;}f = 0.0;for (int j = 0; j < i; j++) {e[j] /= h;f += e[j] * d[j];}double hh = f / (h + h);for (int j = 0; j < i; j++) {e[j] -= hh * d[j];}for (int j = 0; j < i; j++) {f = d[j];g = e[j];for (int k = j; k <= i-1; k++) {V[k][j] -= (f * e[k] + g * d[k]);}d[j] = V[i-1][j];V[i][j] = 0.0;}}d[i] = h;}// Accumulate transformations.for (int i = 0; i < n-1; i++) {V[n-1][i] = V[i][i];V[i][i] = 1.0;double h = d[i+1];if (h != 0.0) {for (int k = 0; k <= i; k++) {d[k] = V[k][i+1] / h;}for (int j = 0; j <= i; j++) {double g = 0.0;for (int k = 0; k <= i; k++) {g += V[k][i+1] * V[k][j];}for (int k = 0; k <= i; k++) {V[k][j] -= g * d[k];}}}for (int k = 0; k <= i; k++) {V[k][i+1] = 0.0;}}for (int j = 0; j < n; j++) {d[j] = V[n-1][j];V[n-1][j] = 0.0;}V[n-1][n-1] = 1.0;e[0] = 0.0;} // Symmetric tridiagonal QL algorithm.private void tql2 () {// This is derived from the Algol procedures tql2, by// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding// Fortran subroutine in EISPACK.for (int i = 1; i < n; i++) {e[i-1] = e[i];}e[n-1] = 0.0;double f = 0.0;double tst1 = 0.0;double eps = Math.pow(2.0,-52.0);for (int l = 0; l < n; l++) {// Find small subdiagonal elementtst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));int m = l;while (m < n) {if (Math.abs(e[m]) <= eps*tst1) {break;}m++;}// If m == l, d[l] is an eigenvalue,// otherwise, iterate.if (m > l) {int iter = 0;do {iter = iter + 1; // (Could check iteration count here.)// Compute implicit shiftdouble g = d[l];double p = (d[l+1] - g) / (2.0 * e[l]);double r = Maths.hypot(p,1.0);if (p < 0) {r = -r;}d[l] = e[l] / (p + r);d[l+1] = e[l] * (p + r);double dl1 = d[l+1];double h = g - d[l];for (int i = l+2; i < n; i++) {d[i] -= h;}f = f + h;// Implicit QL transformation.p = d[m];double c = 1.0;double c2 = c;double c3 = c;double el1 = e[l+1];double s = 0.0;double s2 = 0.0;for (int i = m-1; i >= l; i--) {c3 = c2;c2 = c;s2 = s;g = c * e[i];h = c * p;r = Maths.hypot(p,e[i]);e[i+1] = s * r;s = e[i] / r;c = p / r;p = c * d[i] - s * g;d[i+1] = h + s * (c * g + s * d[i]);// Accumulate transformation.for (int k = 0; k < n; k++) {h = V[k][i+1];V[k][i+1] = s * V[k][i] + c * h;V[k][i] = c * V[k][i] - s * h;}}p = -s * s2 * c3 * el1 * e[l] / dl1;e[l] = s * p;d[l] = c * p;// Check for convergence.} while (Math.abs(e[l]) > eps*tst1);}d[l] = d[l] + f;e[l] = 0.0;}// Sort eigenvalues and corresponding vectors.for (int i = 0; i < n-1; i++) {int k = i;double p = d[i];for (int j = i+1; j < n; j++) {if (d[j] < p) {k = j;p = d[j];}}if (k != i) {d[k] = d[i];d[i] = p;for (int j = 0; j < n; j++) {p = V[j][i];V[j][i] = V[j][k];V[j][k] = p;}}}}// Nonsymmetric reduction to Hessenberg form.private void orthes () {// This is derived from the Algol procedures orthes and ortran,// by Martin and Wilkinson, Handbook for Auto. Comp.,// Vol.ii-Linear Algebra, and the corresponding// Fortran subroutines in EISPACK.int low = 0;int high = n-1;for (int m = low+1; m <= high-1; m++) {// Scale column.double scale = 0.0;for (int i = m; i <= high; i++) {scale = scale + Math.abs(H[i][m-1]);}if (scale != 0.0) {// Compute Householder transformation.double h = 0.0;for (int i = high; i >= m; i--) {ort[i] = H[i][m-1]/scale;h += ort[i] * ort[i];}double g = Math.sqrt(h);if (ort[m] > 0) {g = -g;}h = h - ort[m] * g;ort[m] = ort[m] - g;// Apply Householder similarity transformation// H = (I-u*u'/h)*H*(I-u*u')/h)for (int j = m; j < n; j++) {double f = 0.0;for (int i = high; i >= m; i--) {f += ort[i]*H[i][j];}f = f/h;for (int i = m; i <= high; i++) {H[i][j] -= f*ort[i];}}for (int i = 0; i <= high; i++) {double f = 0.0;for (int j = high; j >= m; j--) {f += ort[j]*H[i][j];}f = f/h;for (int j = m; j <= high; j++) {H[i][j] -= f*ort[j];}}ort[m] = scale*ort[m];H[m][m-1] = scale*g;}}// Accumulate transformations (Algol's ortran).for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {V[i][j] = (i == j ? 1.0 : 0.0);}}for (int m = high-1; m >= low+1; m--) {if (H[m][m-1] != 0.0) {for (int i = m+1; i <= high; i++) {ort[i] = H[i][m-1];}for (int j = m; j <= high; j++) {double g = 0.0;for (int i = m; i <= high; i++) {g += ort[i] * V[i][j];}// Double division avoids possible underflowg = (g / ort[m]) / H[m][m-1];for (int i = m; i <= high; i++) {V[i][j] += g * ort[i];}}}}}// Complex scalar division.private transient double cdivr, cdivi;private void cdiv(double xr, double xi, double yr, double yi) {double r,d;if (Math.abs(yr) > Math.abs(yi)) {r = yi/yr;d = yr + r*yi;cdivr = (xr + r*xi)/d;cdivi = (xi - r*xr)/d;} else {r = yr/yi;d = yi + r*yr;cdivr = (r*xr + xi)/d;cdivi = (r*xi - xr)/d;}}// Nonsymmetric reduction from Hessenberg to real Schur form.private void hqr2 () {// This is derived from the Algol procedure hqr2,// by Martin and Wilkinson, Handbook for Auto. Comp.,// Vol.ii-Linear Algebra, and the corresponding// Fortran subroutine in EISPACK.// Initializeint nn = this.n;int n = nn-1;int low = 0;int high = nn-1;double eps = Math.pow(2.0,-52.0);double exshift = 0.0;double p=0,q=0,r=0,s=0,z=0,t,w,x,y;// Store roots isolated by balanc and compute matrix normdouble norm = 0.0;for (int i = 0; i < nn; i++) {if (i < low | i > high) {d[i] = H[i][i];e[i] = 0.0;}for (int j = Math.max(i-1,0); j < nn; j++) {norm = norm + Math.abs(H[i][j]);}}// Outer loop over eigenvalue indexint iter = 0;while (n >= low) {// Look for single small sub-diagonal elementint l = n;while (l > low) {s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);if (s == 0.0) {s = norm;}if (Math.abs(H[l][l-1]) < eps * s) {break;}l--;}// Check for convergence// One root foundif (l == n) {H[n][n] = H[n][n] + exshift;d[n] = H[n][n];e[n] = 0.0;n--;iter = 0;// Two roots found} else if (l == n-1) {w = H[n][n-1] * H[n-1][n];p = (H[n-1][n-1] - H[n][n]) / 2.0;q = p * p + w;z = Math.sqrt(Math.abs(q));H[n][n] = H[n][n] + exshift;H[n-1][n-1] = H[n-1][n-1] + exshift;x = H[n][n];// Real pairif (q >= 0) {if (p >= 0) {z = p + z;} else {z = p - z;}d[n-1] = x + z;d[n] = d[n-1];if (z != 0.0) {d[n] = x - w / z;}e[n-1] = 0.0;e[n] = 0.0;x = H[n][n-1];s = Math.abs(x) + Math.abs(z);p = x / s;q = z / s;r = Math.sqrt(p * p+q * q);p = p / r;q = q / r;// Row modificationfor (int j = n-1; j < nn; j++) {z = H[n-1][j];H[n-1][j] = q * z + p * H[n][j];H[n][j] = q * H[n][j] - p * z;}// Column modificationfor (int i = 0; i <= n; i++) {z = H[i][n-1];H[i][n-1] = q * z + p * H[i][n];H[i][n] = q * H[i][n] - p * z;}// Accumulate transformationsfor (int i = low; i <= high; i++) {z = V[i][n-1];V[i][n-1] = q * z + p * V[i][n];V[i][n] = q * V[i][n] - p * z;}// Complex pair} else {d[n-1] = x + p;d[n] = x + p;e[n-1] = z;e[n] = -z;}n = n - 2;iter = 0;// No convergence yet} else {// Form shiftx = H[n][n];y = 0.0;w = 0.0;if (l < n) {y = H[n-1][n-1];w = H[n][n-1] * H[n-1][n];}// Wilkinson's original ad hoc shiftif (iter == 10) {exshift += x;for (int i = low; i <= n; i++) {H[i][i] -= x;}s = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-2]);x = y = 0.75 * s;w = -0.4375 * s * s;}// MATLAB's new ad hoc shiftif (iter == 30) {s = (y - x) / 2.0;s = s * s + w;if (s > 0) {s = Math.sqrt(s);if (y < x) {s = -s;}s = x - w / ((y - x) / 2.0 + s);for (int i = low; i <= n; i++) {H[i][i] -= s;}exshift += s;x = y = w = 0.964;}}iter = iter + 1; // (Could check iteration count here.)// Look for two consecutive small sub-diagonal elementsint m = n-2;while (m >= l) {z = H[m][m];r = x - z;s = y - z;p = (r * s - w) / H[m+1][m] + H[m][m+1];q = H[m+1][m+1] - z - r - s;r = H[m+2][m+1];s = Math.abs(p) + Math.abs(q) + Math.abs(r);p = p / s;q = q / s;r = r / s;if (m == l) {break;}if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +Math.abs(H[m+1][m+1])))) {break;}m--;}for (int i = m+2; i <= n; i++) {H[i][i-2] = 0.0;if (i > m+2) {H[i][i-3] = 0.0;}}// Double QR step involving rows l:n and columns m:nfor (int k = m; k <= n-1; k++) {boolean notlast = (k != n-1);if (k != m) {p = H[k][k-1];q = H[k+1][k-1];r = (notlast ? H[k+2][k-1] : 0.0);x = Math.abs(p) + Math.abs(q) + Math.abs(r);if (x == 0.0) {continue;}p = p / x;q = q / x;r = r / x;}s = Math.sqrt(p * p + q * q + r * r);if (p < 0) {s = -s;}if (s != 0) {if (k != m) {H[k][k-1] = -s * x;} else if (l != m) {H[k][k-1] = -H[k][k-1];}p = p + s;x = p / s;y = q / s;z = r / s;q = q / p;r = r / p;// Row modificationfor (int j = k; j < nn; j++) {p = H[k][j] + q * H[k+1][j];if (notlast) {p = p + r * H[k+2][j];H[k+2][j] = H[k+2][j] - p * z;}H[k][j] = H[k][j] - p * x;H[k+1][j] = H[k+1][j] - p * y;}// Column modificationfor (int i = 0; i <= Math.min(n,k+3); i++) {p = x * H[i][k] + y * H[i][k+1];if (notlast) {p = p + z * H[i][k+2];H[i][k+2] = H[i][k+2] - p * r;}H[i][k] = H[i][k] - p;H[i][k+1] = H[i][k+1] - p * q;}// Accumulate transformationsfor (int i = low; i <= high; i++) {p = x * V[i][k] + y * V[i][k+1];if (notlast) {p = p + z * V[i][k+2];V[i][k+2] = V[i][k+2] - p * r;}V[i][k] = V[i][k] - p;V[i][k+1] = V[i][k+1] - p * q;}} // (s != 0)} // k loop} // check convergence} // while (n >= low)// Backsubstitute to find vectors of upper triangular formif (norm == 0.0) {return;}for (n = nn-1; n >= 0; n--) {p = d[n];q = e[n];// Real vectorif (q == 0) {int l = n;H[n][n] = 1.0;for (int i = n-1; i >= 0; i--) {w = H[i][i] - p;r = 0.0;for (int j = l; j <= n; j++) {r = r + H[i][j] * H[j][n];}if (e[i] < 0.0) {z = w;s = r;} else {l = i;if (e[i] == 0.0) {if (w != 0.0) {H[i][n] = -r / w;} else {H[i][n] = -r / (eps * norm);}// Solve real equations} else {x = H[i][i+1];y = H[i+1][i];q = (d[i] - p) * (d[i] - p) + e[i] * e[i];t = (x * s - z * r) / q;H[i][n] = t;if (Math.abs(x) > Math.abs(z)) {H[i+1][n] = (-r - w * t) / x;} else {H[i+1][n] = (-s - y * t) / z;}}// Overflow controlt = Math.abs(H[i][n]);if ((eps * t) * t > 1) {for (int j = i; j <= n; j++) {H[j][n] = H[j][n] / t;}}}}// Complex vector} else if (q < 0) {int l = n-1;// Last vector component imaginary so matrix is triangularif (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {H[n-1][n-1] = q / H[n][n-1];H[n-1][n] = -(H[n][n] - p) / H[n][n-1];} else {cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);H[n-1][n-1] = cdivr;H[n-1][n] = cdivi;}H[n][n-1] = 0.0;H[n][n] = 1.0;for (int i = n-2; i >= 0; i--) {double ra,sa,vr,vi;ra = 0.0;sa = 0.0;for (int j = l; j <= n; j++) {ra = ra + H[i][j] * H[j][n-1];sa = sa + H[i][j] * H[j][n];}w = H[i][i] - p;if (e[i] < 0.0) {z = w;r = ra;s = sa;} else {l = i;if (e[i] == 0) {cdiv(-ra,-sa,w,q);H[i][n-1] = cdivr;H[i][n] = cdivi;} else {// Solve complex equationsx = H[i][i+1];y = H[i+1][i];vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;vi = (d[i] - p) * 2.0 * q;if (vr == 0.0 & vi == 0.0) {vr = eps * norm * (Math.abs(w) + Math.abs(q) +Math.abs(x) + Math.abs(y) + Math.abs(z));}cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);H[i][n-1] = cdivr;H[i][n] = cdivi;if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;} else {cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);H[i+1][n-1] = cdivr;H[i+1][n] = cdivi;}}// Overflow controlt = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));if ((eps * t) * t > 1) {for (int j = i; j <= n; j++) {H[j][n-1] = H[j][n-1] / t;H[j][n] = H[j][n] / t;}}}}}}// Vectors of isolated rootsfor (int i = 0; i < nn; i++) {if (i < low | i > high) {for (int j = i; j < nn; j++) {V[i][j] = H[i][j];}}}// Back transformation to get eigenvectors of original matrixfor (int j = nn-1; j >= low; j--) {for (int i = low; i <= high; i++) {z = 0.0;for (int k = low; k <= Math.min(j,high); k++) {z = z + V[i][k] * H[k][j];}V[i][j] = z;}}}/* ------------------------Constructor* ------------------------ *//** Check for symmetry, then construct the eigenvalue decompositionStructure to access D and V.@param Arg Square matrix*/public EigenvalueDecomposition (Matrix Arg) {double[][] A = Arg.getArray();n = Arg.getColumnDimension();V = new double[n][n];d = new double[n];e = new double[n];issymmetric = true;for (int j = 0; (j < n) & issymmetric; j++) {for (int i = 0; (i < n) & issymmetric; i++) {issymmetric = (A[i][j] == A[j][i]);}}if (issymmetric) {for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {V[i][j] = A[i][j];}}// Tridiagonalize.tred2();// Diagonalize.tql2();} else {H = new double[n][n];ort = new double[n];for (int j = 0; j < n; j++) {for (int i = 0; i < n; i++) {H[i][j] = A[i][j];}}// Reduce to Hessenberg form.orthes();// Reduce Hessenberg to real Schur form.hqr2();}}/* ------------------------Public Methods* ------------------------ *//** Return the eigenvector matrix@return V*/public Matrix getV () {return new Matrix(V,n,n);}/** Return the real parts of the eigenvalues@return real(diag(D))*/public double[] getRealEigenvalues () {return d;}/** Return the imaginary parts of the eigenvalues@return imag(diag(D))*/public double[] getImagEigenvalues () {return e;}/** Return the block diagonal eigenvalue matrix@return D*/public Matrix getD () {Matrix X = new Matrix(n,n);double[][] D = X.getArray();for (int i = 0; i < n; i++) {for (int j = 0; j < n; j++) {D[i][j] = 0.0;}D[i][i] = d[i];if (e[i] > 0) {D[i][i+1] = e[i];} else if (e[i] < 0) {D[i][i-1] = e[i];}}return X;}private static final long serialVersionUID = 1; }

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