QR算法求解矩阵特征值、特征向量(Java代码)
矩阵求特征值特征向量Java代码
- 1.代码介绍
- 2.参考文章
1.代码介绍
代码中实现的模块如下：
- 矩阵转置
- 点乘
- 自动索引生成
- 矩阵复制
- 获取给定索引矩阵所在行
- 矩阵行列式
- 矩阵正交化
- 矩阵求逆
- 矩阵QR分解(注：QR分解函数只返回一个矩阵，设输入矩阵为m×n，则前m行为Q矩阵，后n行为R矩阵)
- 矩阵乘法
- 矩阵相减
- 求特征值(注：返回值为二维矩阵，对角线上的方为特征值)
- 求特征向量
所有代码如下：
注意：本文中代码所在包为util，类名为Eig，使用时注意对于及修改；代码传入参数等也请注意阅读函数说明。
package util;
import java.util.Arrays;
/**
* @(#)Eig.java The class of Eig.
* @author: 因吉
* @Email: [email protected]
* @Created: November 23, 2019.
* @Last_modified: November 23, 2019.
*/
public class Eig {
/**
************
* Array transpose.
*
* @param paraArray The transposing array(double[][]).
*
* @return returnArray The transposed array(double[][]).
************
*/
public static double[][] arrayTranspose(double[][] paraArray) {
int tempM = paraArray.length;
int tempN = paraArray[0].length;
double[][] returnArray = new double[tempN][tempM];
for (int i = 0; i < tempM; i++) {
for (int j = 0; j < tempN; j++) {
returnArray[j][i] = paraArray[i][j];
} // Of for j
} // Of for i
return returnArray;
}// Of arrayTranspose
/**
************
* Array corresponding elements multiply and add all elements.点积
*
* @param paraFirstArray The first array(double[]).
* @param paraSecondArray The second array(double[]).
* @return The result(double).
************
*/
public static double arrayMultiplyAndAdd(double[] paraFirstArray, double[] paraSecondArray) {
int tempM = paraFirstArray.length;
double resultMultipliedArray = 0;
for (int i = 0; i < tempM; i++) {
resultMultipliedArray += paraFirstArray[i] * paraSecondArray[i];
} // Of for i
return resultMultipliedArray;
}// Of arrayMultiplyAndAdd
/**
************
* Get the index of array.
*
* @param paraLen The length of array(int).
* @return The index of array(int[]).
************
*/
public static int[] arrayIndexAuto(int paraLen) {
int[] returnArray = new int[paraLen];
for (int i = 0; i < paraLen; i++) {
returnArray[i] = i;
}
return returnArray;
}// Of arrayIndexAuto
/**
************
* Get the index of array.
*
* @param paraStrat The start index(int).
* @param paraEnd The end index(int).
* @return The index of array(int[]).
************
*/
public static int[] arrayIndexAuto(int paraStrat, int paraEnd) {
int[] returnArray = new int[paraEnd - paraStrat];
for (int i = 0; i < returnArray.length; i++) {
returnArray[i] = i + paraStrat;
}
return returnArray;
}// Of arrayIndexAuto
/**
************
* Copy array.
*
* @param paraArray The array(double[][]).
* @return The copied array(double[][]).
************
*/
public static double[][] arrayCopy(double[][] paraArray) {
int tempM = paraArray.length;
double[][] resultArray = new double[tempM][];
for (int i = 0; i < tempM; i++) {
resultArray[i] = paraArray[i];
} // Of for i
return resultArray;
}// Of arrayCopy
/**
************
* Get the elements of the corresponding row of the incoming array.
*
* @param paraArray The import array(double[][]).
* @param paraIndex The index i(int[]).
* @return The row elements corresponding incoming array(doubel[][]).
************
*/
public static double[][] arrayRowValue(double[][] paraArray, int[] paraIndex) {
int tempParaIndex = paraIndex.length;
double[][] returnRowValue = new double[tempParaIndex][];
for (int i = 0; i < returnRowValue.length; i++) {
returnRowValue[i] = paraArray[paraIndex[i]];
} // Of for i
return returnRowValue;
}// Of arrayRowValue
/**
************
* The determinant of matrix.求解矩阵行列式
*
* @param paraMatrix The given matrix(double[][]).
*
* @return The answer of determinant.
************
*/
public static double matrixDeterminant(double[][] paraMatrix) {
int tempM = paraMatrix.length;
int tempN = paraMatrix[0].length;
if (tempM == 1) {
return paraMatrix[0][0];
} else if (tempM == 2) {
return paraMatrix[0][0] * paraMatrix[1][1] - paraMatrix[1][0] * paraMatrix[0][1];
} // Of if
double resultValue = 0;
int k;
for (int i = 0; i < tempN; i++) {
double[][] tempMatrix = new double[tempM - 1][tempN - 1];
for (int j = 0; j < tempM - 1; j++) {
k = 0;
while (k < tempN - 1) {
if (k < i) {
tempMatrix[j][k] = paraMatrix[j + 1][k];
} else {
tempMatrix[j][k] = paraMatrix[j + 1][k + 1];
} // Of if
k++;
} // Of while
} // Of for j
resultValue += (Math.pow(-1., i)) * paraMatrix[0][i] * matrixDeterminant(tempMatrix);
} // Of for i
return resultValue;
}// Of matrixDeterminant
/**
************
* The gram schimidt of matrix.求解矩阵正交化
*
* @param paraMatrix The given matrix(double[][]).
*
* @return The gram schimidt of matrix.
************
*/
public static double[][] matrixGramSchimidt(double[][] paraMatrix) {
if (paraMatrix == null) {
return null;
} // Of if
double[][] tempTransposedMatrix = arrayTranspose(paraMatrix);
int tempM = tempTransposedMatrix.length;
int tempN = tempTransposedMatrix[0].length;
double[][] resultMatrix = new double[tempM][tempN];
double tempValue = 0;
double tempFactor = 0;
for (int i = 0; i < tempM; i++) {
for (int j = 0; j < tempN; j++) {
tempValue = tempTransposedMatrix[i][j];
for (int k = 0; k < i; k++) {
tempFactor = (1. * arrayMultiplyAndAdd(tempTransposedMatrix[i], resultMatrix[k]))
/ arrayMultiplyAndAdd(resultMatrix[k], resultMatrix[k]);
tempValue -= tempFactor * resultMatrix[k][j];
} // Of for k
resultMatrix[i][j] = tempValue;
} // Of for j
} // Of for i
return arrayTranspose(resultMatrix);
}// Of matrixGramSchimidt
/**
************
* The inverse of matrix.矩阵求逆
*
* @param paraMatrix The given matrix(double[][]).
*
* @return The inverse of matrix(double[][]).
************
*/
public static double[][] matrixInverse(double[][] paraMatrix) {
if (paraMatrix == null) {
return null;
} // Of if
// Make sure that this matrix is invertible.
if (matrixDeterminant(paraMatrix) == 0) {
System.out.println("Fetal error, the matrix can not be invertible.");
System.exit(0);
} // Of if
int tempM = paraMatrix.length;
int tempN = paraMatrix[0].length * 2;
double[][] tempCopyMatrix = new double[tempM][tempN];
for (int i = 0; i < tempM; i++) {
for (int j = 0; j < tempN; j++) {
if (j < tempM) {
tempCopyMatrix[i][j] = paraMatrix[i][j];
} else {
if (j == i + tempM) {
tempCopyMatrix[i][j] = 1;
} // Of if
} // Of if
} // Of for j
} // Of for i
double tempTimes = 0;
for (int i = 0; i < tempM; i++) {
if (tempCopyMatrix[i][i] == 0) {
int j;
for (j = i + 1; j < tempM; j++) {
if (tempCopyMatrix[j][i] != 0) {
break;
} // Of if
} // Of for j
if (j != i + 1) {
for (int k = 0; k < tempN; k++) {
double temppVlaue = tempCopyMatrix[i][k];
tempCopyMatrix[i][k] = tempCopyMatrix[j][k];
tempCopyMatrix[j][k] = temppVlaue;
} // Of for k
} // Of if
} // Of if
for (int j = i + 1; j < tempM; j++) {
if (tempCopyMatrix[j][i] != 0) {
tempTimes = (tempCopyMatrix[j][i]) / tempCopyMatrix[i][i];
for (int k = i; k < tempN; k++) {
tempCopyMatrix[j][k] /= tempTimes;
tempCopyMatrix[j][k] -= tempCopyMatrix[i][k];
} // Of for k
} // Of if
} // Of for j
} // Of for i
for (int i = 0; i < tempM; i++) {
for (int j = i + 1; j < tempN / 2.; j++) {
if (tempCopyMatrix[i][j] != 0) {
tempTimes = tempCopyMatrix[i][j] / tempCopyMatrix[j][j];
for (int k = j; k < tempN; k++) {
tempCopyMatrix[i][k] -= tempTimes * tempCopyMatrix[j][k];
} // Of for k
} // Of if
} // Of for j
} // Of for i
for (int i = 0; i < tempM; i++) {
tempTimes = tempCopyMatrix[i][i];
for (int j = 0; j < tempN; j++) {
tempCopyMatrix[i][j] /= tempTimes;
} // Of for j
} // Of for i
double[][] resultMatrix = new double[tempM][tempN / 2];
for (int i = 0; i < tempM; i++) {
for (int j = 0; j < tempN / 2; j++) {
resultMatrix[i][j] = tempCopyMatrix[i][j + tempN / 2];
} // Of for j
} // Of for i
return resultMatrix;
}// Of matrixInverse
/**
************
* The QR-decomposition of matrix.矩阵QR分解
*
* @param paraMatrix The given matrix(double[][]).
* @param paraMatrixQ The given matrix, the size echo paraMatrix(double[][]).
************
*/
public static double[][] matrixQrDecomposition(double[][] paraMatrix) {
double[][] tempOrthogonalMatrix = arrayTranspose(matrixGramSchimidt(paraMatrix));
int tempM = tempOrthogonalMatrix.length;
int tempN = tempOrthogonalMatrix[0].length;
double[][] tempMatrixQ = new double[tempM][tempN];
for (int i = 0; i < tempM; i++) {
double tempMag = magnitude(tempOrthogonalMatrix[i]);
for (int j = 0; j < tempN; j++) {
tempMatrixQ[i][j] = tempOrthogonalMatrix[i][j] / tempMag;
} // Of for j
} // Of for i
double[][] tempMatrixR = matrixMultiply(tempMatrixQ, paraMatrix);
double[][] resultSummary = new double[tempM + tempN][tempM];
for (int i = 0; i < tempN; i++) {
for (int j = 0; j < tempM; j++) {
resultSummary[i][j] = tempMatrixQ[j][i];
} // Of for j
} // Of for i
for (int i = tempN; i < resultSummary.length; i++) {
for (int j = 0; j < tempM; j++) {
resultSummary[i][j] = tempMatrixR[i - tempN][j];
} // Of for j
} // Of for i
return resultSummary;
}// Of matrixQrDecomposition
/**
************
* The magnitude.
*
* @param paraMatrix The given matrix(double[]).
* @return The answer.
************
*/
private static double magnitude(double[] paraMatrix) {
return Math.sqrt(arrayMultiplyAndAdd(paraMatrix, paraMatrix));
}// Of magnitude
/**
************
* The matrix multiply.
*
* @param paraFirstMatrix The given first matrix(double[][]).
* @param paraSecondMatrix The given second matrix(double[][]).
* @return The answer(double[][]).
************
*/
public static double[][] matrixMultiply(double[][] paraFirstMatrix, double[][] paraSecondMatrix) {
if (paraFirstMatrix == null || paraSecondMatrix == null) {
return null;
} // Of if
if (paraFirstMatrix[0].length != paraSecondMatrix.length) {
System.out.println("Fetal error: The size of two inputed matrix is illegally.");
System.exit(0);
} // Of if
int tempMa = paraFirstMatrix.length;
int tempNa = paraFirstMatrix[0].length;
int tempNb = paraSecondMatrix[0].length;
double[][] resultMatrix = new double[tempMa][tempNb];
for (int i = 0; i < tempMa; i++) {
for (int j = 0; j < tempNb; j++) {
double tempSum = 0;
for (int k = 0; k < tempNa; k++) {
tempSum += paraFirstMatrix[i][k] * paraSecondMatrix[k][j];
} // Of for k
resultMatrix[i][j] = tempSum;
} // Of for j
} // Of for i
return resultMatrix;
}// Of matrixMultiply
/**
************
* The eig of matrix.
*
* @param paraMatrix The given matrix(double[][]).
* @param paraIter The maximum iteration.
* @return The eig of matrix.
************
*/
public static double[][] matrixEigValue(double[][] paraMatrix, int paraIter) {
int tempM = paraMatrix.length;
int tempN = paraMatrix[0].length;
int[] tempIndexQ = arrayIndexAuto(tempM);
int[] tempIndexR = arrayIndexAuto(tempM, tempM + tempN);
for (int i = 0; i < paraIter; i++) {
double[][] tempSummary = matrixQrDecomposition(paraMatrix);
double[][] tempMatrixQ = arrayRowValue(tempSummary, tempIndexQ);
double[][] tempMatrixR = arrayRowValue(tempSummary, tempIndexR);
paraMatrix = matrixMultiply(tempMatrixR, tempMatrixQ);
} // Of for i
return paraMatrix;
}// Of matrixEigValue
/**
************
* The eig vector of matrix.求矩阵特征向量
*
* @param paraMatrix The given matrix(double[][]).
* @param paraIter The maximum iteration.
* @return The eig vector of matrix.
************
*/
public static double[][] matrixEigVector(double[][] paraMatrix, int paraIter) {
int tempM = paraMatrix.length;
int tempN = paraMatrix[0].length;
double[][] tempMatrix = matrixEigValue(arrayCopy(paraMatrix), paraIter);
for (int i = 0; i < tempM; i++) {
for (int j = 0; j < tempN; j++) {
if (i != j) {
tempMatrix[i][j] = 0;
}//Of if
}//Of for j
}//Of for i
return matrixInverse(matrixSubtract(paraMatrix, tempMatrix));
}//Of matrixEigVector
/**
************
* The subtract of matrix.求矩阵相减
*
* @param paraFirstMatrix The first given matrix(double[][]).
* @param paraFirstMatrix The second given matrix(double[][]).
* @return The subtract of two matrix.
************
*/
public static double[][] matrixSubtract(double[][] paraFirstMatrix, double[][] paraSecondMatrix) {
if (paraFirstMatrix == null || paraSecondMatrix == null) {
return null;
}//Of if
int tempMFirst = paraFirstMatrix.length;
int tempNFirst = paraFirstMatrix[0].length;
int tempMSecond = paraSecondMatrix.length;
int tempNSecond = paraSecondMatrix[0].length;
if (tempMFirst != tempMSecond || tempNFirst != tempNSecond) {
System.out.println("Fetal error, the inputed matrix is illegal in matrixSub(double[][], double[][])");
System.exit(0);
}//Of if
double[][] resultMatrix = new double[tempMFirst][tempNFirst];
for (int i = 0; i < tempMFirst; i++) {
for (int j = 0; j < tempNFirst; j++) {
resultMatrix[i][j] = paraFirstMatrix[i][j] - paraSecondMatrix[i][j];
}//Of for j
}//Of for i
return resultMatrix;
}//Of matrixSubtract
/**
************
* The main
************
*/
public static void main(String[] args) {
// double[][] tempData = { { 1, 0, 0 }, { 1, 1, 0 }, { 1, 1, 1 } };
double[][] tempData = {{2,1}, {1,2}};
double[][] tempEigValue = matrixEigValue(tempData, 1000);
double[][] tempEigVector = matrixEigVector(tempData, 1000);
System.out.println("The eig value is: " + Arrays.deepToString(tempEigValue));
System.out.println("The eig vector is: " +Arrays.deepToString(tempEigVector));
}// Of main
}//Of Eig
2.参考文章
- QR算法讲解
- 矩阵求逆、正交化等