I need to multiply two matrices. I understand pretty well how matrices work however in Java I am finding this a bit complex, so I researched a bit and found this.
public static int[][] multiply(int a[][], int b[][]) {
int aRows = a.length,
aColumns = a[0].length,
bRows = b.length,
bColumns = b[0].length;
int[][] resultant = new int[aRows][bColumns];
for(int i = 0; i < aRows; i++) { // aRow
for(int j = 0; j < bColumns; j++) { // bColumn
for(int k = 0; k < aColumns; k++) { // aColumn
resultant[i][j] += a[i][k] * b[k][j];
}
}
}
return resultant;
This code works fine. However the problem with this is that I need to multiply a single dimension matrix (1*5) by a multidimensional matrix (5*4), so the result will be (1*4) matrix and later on in the same program multiply a (1*4) matrix by a (4*3) matrix resulting in (1*3).
And I need to store the single dimension matrix in a normal array (double []) not multidimensional one!
I altered this code to the following but it still doesn't resolve the correct results.
public static double[] multiplyMatrices(double[] A, double[][] B) {
int xA = A.length;
int yB = B[0].length;
double[] C = new double[yB];
for (int i = 0; i < yB; i++) { // bColumn
for (int j = 0; j < xA; j++) { // aColumn
C[i] += A[j] * B[j][i];
}
}
return C;
Thanks in advance for any tips you may give :)
You can use RealMatrix to make it easier.
RealMatrix result = MatrixUtils.createRealMatrix(a).multiply(MatrixUtils.createRealMatrix(b));
double[] array = result.getRow(0);
Related
I'm not sure whether this is a maths.se or a SO question, but I'm going with SO as I think it's related to my Java.
I'm following a text book on Gaussian Processes (R&W) and implementing some examples in Java. One common step for several examples is to generate a Cholesky decomposition of a covariance matrix. In my attempt I can get successful results for matrices up to a limited size (33x33). However, for anything larger a NaN appears in the diagonal (at 32,32) and so all subsequent values in the matrix are likewise NaNs.
The code is shown below, and the source of the NaN is indicated in the cholesky method. Essentially the covariance element a[32][32] is 1.0, but the value of sum is a little over this (1.0000001423291431), so the square root is imaginary. So my questions are:
Is this an expected result from linear algebra, or, e.g., an
artefact of my implementation?
How is this problem best avoided in practice?
Note that I'm not looking for recommendations of libraries to use. This is simply for my own understanding.
Apologies for the length, but I've tried to provide a complete MWE:
import static org.junit.Assert.assertFalse;
import org.junit.Test;
public class CholeskyTest {
#Test
public void testCovCholesky() {
final int n = 34; // Test passes for n<34
final double[] xData = getSpread(-5, 5, n);
double[][] cov = covarianceSE(xData);
double[][] lower = cholesky(cov);
for(int i=0; i<n; ++i) {
for(int j=0; j<n; ++j) {
assertFalse("NaN at " + i + "," + j, Double.isNaN(lower[i][j]));
}
}
}
/**
* Generate n evenly space values from min to max inclusive
*/
private static double[] getSpread(final double min, final double max, final int n) {
final double[] values = new double[n];
final double delta = (max - min)/(n - 1);
for(int i=0; i<n; ++i) {
values[i] = min + i*delta;
}
return values;
}
/**
* Calculate the covariance matrix for the given observations using
* the squared exponential (SE) covariance function.
*/
private static double[][] covarianceSE (double[] v) {
final int m = v.length;
double[][] k = new double[m][];
for(int i=0; i<m; ++i) {
double vi = v[i];
double row[] = new double[m];
for(int j=0; j<m; ++j) {
double dist = vi - v[j];
row[j] = Math.exp(-0.5*dist*dist);
}
k[i] = row;
}
return k;
}
/**
* Calculate lower triangular matrix L such that LL^T = A
* Using Cholesky decomposition from
* https://rosettacode.org/wiki/Cholesky_decomposition#Java
*/
private static double[][] cholesky(double[][] a) {
final int m = a.length;
double[][] l = new double[m][m];
for(int i = 0; i< m;i++){
for(int k = 0; k < (i+1); k++){
double sum = 0;
for(int j = 0; j < k; j++){
sum += l[i][j] * l[k][j];
}
l[i][k] = (i == k) ? Math.sqrt(a[i][i] - sum) : // Source of NaN at 32,32
(1.0 / l[k][k] * (a[i][k] - sum));
}
}
return l;
}
}
Hmm, I think I've found an answer to my own question, from the same textbook I was following. From R&W p.201:
In practice it may be necessary to add a small multiple of the
identity matrix $\epsilon I$ to the covariance matrix for numerical
reasons. This is because the eigenvalues of the matrix K can decay
very rapidly [...] and without this stabilization the Cholesky
decomposition fails. The effect on the generated samples is to add
additional independent noise of variance $epsilon$.
So the following change seems to be sufficient:
private static double[][] cholesky(double[][] a) {
final int m = a.length;
double epsilon = 0.000001; // Small extra noise value
double[][] l = new double[m][m];
for(int i = 0; i< m;i++){
for(int k = 0; k < (i+1); k++){
double sum = 0;
for(int j = 0; j < k; j++){
sum += l[i][j] * l[k][j];
}
l[i][k] = (i == k) ? Math.sqrt(a[i][i]+epsilon - sum) : // Add noise to diagonal values
(1.0 / l[k][k] * (a[i][k] - sum));
}
}
return l;
}
I just finished writing my own version of a Cholesky Decomposition routine in C++ and JavaScript. Instead of computing L, it computes U, but I would be curious to test it with the matrix that causes the NaN error. Would you be able to post the matrix here, or contact me (info in Profile.)
I have a 2-d array of double numbers it's 48 by 48, I am trying to make a method that will allow the user to select a specific amount e.g. = 7 by 7 and then put that into a new 2d array.
public static double[][] amountOTP(double [][] a, int x){
a = new double[x][x];
return a;
}
thats all i have right now, this takes an 2d array as input however even though i specified x it doesn't work.
When you want to cut it ro a smaller size and copy the part of the original array, this should work:
public static double [][] cutArray (double [][] a, int newSize){
if (x > a.length)
throw new IllegalArgumentException ("Can only make array smaller");
double [][] b = new double [newSize][newSize];
for (int i = 0; i < newSize; i++){
for (int j = 0; j < newSize; j++){
b [i][j] = a [i][j];
}
}
return b;
}
The solution below considers situations in which the requested new two-dimensional array length is greater than the original in which case we simply return the original.
Example:
public static double[][] amountOTP(double [][] a, int x){
if(x > a.length) return a;
for (double[] arr : a)
if(arr.length < x)
return a;
double[][] newArray = new double[x][x];
for (int i = 0; i < x; i++)
for (int j = 0; j < x; j++)
newArray[i][j] = a[i][j];
return newArray;
}
Sorry if I'm wrong, but I'm assuming for now that you want your method to return a smaller 2d array which contains some of the values of the given array, in which case you would need to change your method to this:
public static double[][] amountOTP(double [][] a, int x) {
double[][] b = new double[x][x];
x = Math.min(x, a.length);
for(int i = 0;i < x; i++)
for(int j = 0;j < x; j++)
b[i][j] = a[i][j];
return b;
}
This should work fine but feel free to comment and inform me if I left anything out or it doesn't work; I'm here to help. Anyways, I hope this works for you. (Also, if this isn't the answer you were looking for, feel free to tell me.)
Note: This should avoid IndexOutOfBounds exceptions, so it will still work correctly if the user gives an x value bigger than the size of a. The 2d array that this method returns will just have some values of zero where it couldn't find any numbers.
I think it's something like that you're looking for :
public static double[][] amountOTP(double [][] a, int x){
double [][] ret = new double[x][x];
for(int i = 0; i < x; i++)
for(int j = 0; j < x; j++)
ret[i][j] = a[i][j];
return ret;
}
But you have to be careful with the parameters because it can cause an IndexOutOfBounds exception
I see you already have a lot of answers, but nobody was making use of the excellent Arrays.copyOf Java API method:
public static double[][] amountOTP(double [][] a, int x){
a = Arrays.copyOf(a, x);
for(int i=0; i<a.length; i++) {
if(a[i] != null) {
a[i] = Arrays.copyOf(a[i], x);
} else {
a[i] = new double[x]; // allows growth
}
}
return a;
}
you are actually not yet initializing the array so the method should be like this
public static void main(String[] args)
{
double[][] a=amountOTP(3);
for(int j=0;j<a.length;++j)
{
for(int i=0;i<a[j].length;++i)
{
a[j][i]=2;
}
}
for(int j=0;j<a.length;++j)
{
for(int i=0;i<a[j].length;++i)
{
System.out.println(a[j][i]);
}
}
}
public static double[][] amountOTP(int x)
{
return new double[x][x];
}
I have written a series of matrix operations where I take a 2 dimensional float array, treat it as a matrix, and perform matrix operations on it to acquire an inverse. My problem has been that although the array I am using with the class methods isn't part of the class, every time I run the method with the array as the parameter, the array itself also becomes modified.
First I will describe how I got the inverse of my matrix and then I will show the output.
The steps to taking the inverse of a matrix are as follows:
Get the cofactor matrix (i.e. create a matrix of matrix minors of the original matrix and then negate every other entry. If C = Cofactor Matrix, M = Matrix of Minors, i is the current row, and j is the current column, then C[ i ][ j ] = M[ i ][ j ]*( -1 )^( i + j )
Convert the cofactor matrix to the adjugate (also known as adjoint) matrix by transposing (replacing row, column entry by its analogous column, row entry and vice versa) the cofactor matrix. If C = Cofactor Matrix, A = Adjugate Matrix, i is the current row, and j is the current column, then A[ i ][ j ] = C[ j ][ i ]
Finally, take one over determinant of the original matrix and multiply the adjugate matrix by that value. If I = Inverse Matrix, A = Adjugate Matrix and D = Determinant, then I = (1/D)*A
In order to test if you have truly acquired the Matrix Inverse of a Matrix, one can multiply the original matrix by its inverse to get the identity matrix.
If I = Inverse, O = Original Matrix, and id = Identity Matrix then O*I = id
Now I will present the code where I implement these operations. For the sake of conciseness, I will not describe how to get the Matrix of Minors or the Determinant, but the problem I have been encountering will become apparent anyways.
public class MatrixOperations {
//Note: this method works fine. There are no problems.
public float determinant(float [][] a)
{
float [][] temp_mat;
float res = 0;
//assuming a square matrix
/*If it's a 2X2, then use the formula for a determinant of
2X2 matrices.*/
if(a.length == 2)
{
return a[0][0]*a[1][1]-a[0][1]*a[1][0];
}
/*Otherwise do the determinant formula recursively until your
determinant is made up of 2X2 matrix determinants and scalar products*/
else
{
temp_mat = new float[a.length-1][a.length-1];
int placej = 0;
int placei = 0;
for(int k = 0; k<a.length;k++)
{
for(int j = 0; j<a.length; j++)
{
for(int i = 1; i < a.length; i++)
{
placei = i-1;
if(j != k)
{
if(j < k)
{
temp_mat[placei][j] = a[i][j];
}
else if(j > k)
{
if (i == 1){
placej = j-1;
}
temp_mat[placei][placej] = a[i][j];
}
}
}
}
res+=a[0][k]*determinant(temp_mat)*(int)Math.pow(-1, k);
}
return res;
}
}
//Note: this method also works fine
//Scalar product method
public float[][] mul(float[][] m, float r)
{
float[][] res = new float[m.length][m.length];
for(int i = 0; i < m.length; i++)
{
for(int j = 0; j < m.length; j++)
{
res[i][j]= m[i][j]*r;
}
}
return res;
}
//Note: This method also works fine
public float[][] mul(float[][] m,float[][] n)
{
float[][] res = new float[m.length][m.length];
for(int i = 0; i < m.length; i++)
{
for(int j = 0; j < m.length; j++)
{
for(int k = 0; k < m.length; k++)
{
res[i][j] += m[i][k]*m[k][i];
}
}
}
return res;
}
//The method for creating a matrix of minors
//Here I start having problems
public float[][] minor(float [][] m)
{
float [][] minor_mat = new float [m.length][m.length];
//If the matrix is greater than a 2X2, use this to generate a matrix of minors
if(m.length > 2)
{
float [][] current_minor = new float [m.length-1][m.length-1];
int placei = 0;
int placej = 0;
for(int i = 0; i < m.length; i++)
{
for(int j = 0; j < m.length; j++)
{
for(int k = 0; k < m.length; k++)
{
for(int l = 0; l < m.length; l++)
{
if(i != k && j != l)
{
if(k<i)
placei = k;
else if(k>i)
placei = k-1;
if(l<j)
placej = l;
else if(l>j)
placej = l-1;
current_minor[placei][placej] = m[k][l];
}
}
}
minor_mat[i][j] = this.determinant(current_minor);
}
}
}
//otherwise use the definition for 2X2 matrix of minors
else
{
//even though minor_mat is using m.clone() rather than m, when I return the result, m has still been modified for some reason.
minor_mat = m.clone()
float temp;
temp = minor_mat[0][0];
minor_mat[0][0] = minor_mat[1][1];
minor_mat[1][1] = temp;
temp = minor_mat[0][1];
minor_mat[0][1] = minor_mat[1][0];
minor_mat[1][0] = temp;
}
return minor_mat;
}
//the same problem occurs here as it did in the minor method
//m appears to get modified even though I only use m.clone()
public float[][] cofactor(float [][] m)
{
float[][] res = m.clone();
res = this.minor(res)
for(int i = 0; i < m.length; i++)
{
for(int j = 0; j < m.length; j++)
{
res[i][j] = res[i][j]*(int)Math.pow(-1, i + j);
}
}
return res;
}
//The following transpose, adjugate, and inverse methods have the same problem
public float[][] transpose(float[][] m)
{
float[][] res = new float[m.length][m.length];
float temp = 0;
for(int i = 0; i < m.length; i++)
{
for(int j = 0; j < m.length; j++)
{
temp = m[i][j];
res[i][j] = m[j][i];
res[j][i] = temp;
}
}
return res;
}
public float[][] adjugate(float[][] m)
{
float[][] res = this.transpose(this.cofactor(m));
return res;
}
public float[][] inverse(float[][] m)
{
float[][] res = this.mul(this.adjugate(m), (1/this.determinant(m)));
return res;
}
//print out the matrix in square form
public void matrixprint(float [][] m)
{
for(int i = 0; i < m.length; i++)
{
System.out.println("");
for(int j = 0; j < m[i].length; j++){
System.out.print(m[i][j] + " ");
}
}
System.out.println("\n");
}
}
Now the main class and the main method that creates an instance of the MatrixOperations class and uses its methods on a 2X2 matrix.
public class Main {
public static void main(String[] args) {
MatrixOperations mo = new MatrixOperations();
//Create a 2X2 matrix called "matrix" and set its elements
//Then perform each step on "matrix" and finally test if you have acquired the correct inverse
float [][] matrix = new float[2][2];
matrix[0][0] = 2;
matrix [0][1] = 5;
matrix [1][0] = 4;
matrix [1][1] = 3;
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Minor = ");
mo.matrixprint(mo.minor(matrix));
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Cofactor = ");
mo.matrixprint(mo.cofactor(matrix));
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Adjugate = ");
mo.matrixprint(mo.adjugate(matrix));
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Determinant = ");
System.out.println(mo.determinant(matrix));
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Inverse = ");
mo.matrixprint(mo.inverse(matrix));
System.out.println("Matrix = ");
mo.matrixprint(matrix);
System.out.println("Identity = ");
mo.matrixprint(mo.mul(mo.inverse(matrix), matrix));
}
}
Now you will see that when I show the output, every time I use a method on "matrix", and reprint "matrix", "matrix" itself has been modified even though my methods only use a copy of "matrix" and not "matrix" itself.
Output:
Matrix =
2.0 5.0
4.0 3.0
Minor =
3.0 4.0
5.0 2.0
Matrix =
3.0 4.0
5.0 2.0
Cofactor =
3.0 -4.0
-5.0 2.0
Matrix =
3.0 -4.0
-5.0 2.0
Adjugate =
3.0 5.0
4.0 2.0
Matrix =
3.0 4.0
5.0 2.0
Determinant =
-14.0
Matrix =
3.0 4.0
5.0 2.0
Inverse =
-0.21428573 0.35714287
0.2857143 -0.14285715
Matrix =
3.0 -4.0
-5.0 2.0
Identity =
0.1479592 0.1479592
0.12244898 0.12244898
Any help/explanation as to why this happens would be appreciated.
This line does a shallow clone;
float[][] res = m.clone();
This copies the res which is array fo references to arrays. but not any of the arrays res points to. Most likely what you wanted is
float[][] res = new float[m.length][];
for (int i = 0; i < m.length; i++)
res[i] = m[i].clone();
It's because you are passing reference of matrix object in the methods of MatrixOperations class. It's not a copy of matrix object.
From Java doc:
Reference data type parameters, such as objects, are also passed into
methods by value. This means that when the method returns, the
passed-in reference still references the same object as before.
A two-dimensional array is just an array of arrays.
clone() on an array just does a shallow clone.
So you have a new cloned outer array, but it references the same entries (the inner arrays).
After cloning the outer array, iterate over the outer array and clone all inner arrays to get a deep clone.
I'm trying to write a method that calculates the exponential of a square matrix. In this instance, the matrix is a square array of value:
[1 0]
[0 10]
and the method should return a value of:
[e 0]
[0 e^10]
However, when I run my code, I get a range of values depending on what bits I've rearranged, non particularly close to the expected value.
The way the method works is to utilise the power series for the matrix, so basically for a matrix A, n steps and an identity matrix I:
exp(A) = I + A + 1/2!*AA + 1/3!*AAA + ... +1/n!*AAA..
The code follows here. The method where I'm having the issue is the method exponential(Matrix A, int nSteps). The methods involved are enclosed, and the Matrix objects take the arguments (int m, int n) to create an array of size double[m][n].
public static Matrix multiply(Matrix m1, Matrix m2){
if(m1.getN()!=m2.getM()) return null;
Matrix res = new Matrix(m1.getM(), m2.getN());
for(int i = 0; i < m1.getM(); i++){
for(int j = 0; j < m2.getN(); j++){
res.getArray()[i][j] = 0;
for(int k = 0; k < m1.getN(); k++){
res.getArray()[i][j] = res.getArray()[i][j] + m1.getArray()[i][k]*m2.getArray()[k][j];
}
}
}
return res;
}
public static Matrix identityMatrix(int M){
Matrix id = new Matrix(M, M);
for(int i = 0; i < id.getM(); i++){
for(int j = 0; j < id.getN(); j++){
if(i==j) id.getArray()[i][j] = 1;
else id.getArray()[i][j] = 0;
}
}
return id;
}
public static Matrix addMatrix(Matrix m1, Matrix m2){
Matrix m3 = new Matrix(m1.getM(), m2.getN());
for(int i = 0; i < m3.getM(); i++){
for(int j = 0; j < m3.getN(); j++){
m3.getArray()[i][j] = m1.getArray()[i][j] + m2.getArray()[i][j];
}
}
return m3;
}
public static Matrix scaleMatrix(Matrix m, double scale){
Matrix res = new Matrix(m.getM(), m.getN());
for(int i = 0; i < res.getM(); i++){
for(int j = 0; j < res.getN(); j++){
res.getArray()[i][j] = m.getArray()[i][j]*scale;
}
}
return res;
}
public static Matrix exponential(Matrix A, int nSteps){
Matrix runtot = identityMatrix(A.getM());
Matrix sum = identityMatrix(A.getM());
double factorial = 1.0;
for(int i = 1; i <= nSteps; i++){
sum = Matrix.multiply(Matrix.scaleMatrix(sum, factorial), A);
runtot = Matrix.addMatrix(runtot, sum);
factorial /= (double)i;
}
return runtot;
}
So my question is, how should I modify my code, so that I can input a matrix and a number of timesteps to calculate the exponential of said matrix after said timesteps?
My way to go would be to keep two accumulators :
the sum, which is your approximation of exp(A)
the nth term of the series M_n, that is A^n/n!
Note that there is a nice recursive relationship with M_n: M_{n+1} = M_n * A / (n+1)
Which yields :
public static Matrix exponential(Matrix A, int nSteps){
Matrix seriesTerm = identityMatrix(A.getM());
Matrix sum = identityMatrix(A.getM());
for(int i = 1; i <= nSteps; i++){
seriesTerm = Matrix.scaleMatrix(Matrix.multiply(seriesTerm,A),1.0/i);
sum = Matrix.addMatrix(seriesTerm, sum);
}
return sum;
}
I totally understand the sort of thrill that implementing such algorithms can give you. But if this is not a hobby project, I concur that you should that you should use a library for this kind of stuff. Making such computations precise and efficient is really not a trivial matter, and a huge wheel to reinvent.
Basically my HW says to ask user for matrix of A. Then ask user by how much he would like to power Matrix A.
So basically,
I need to find a way to raise a matrix to the power. I can multiply them, but it's harder to raise them to the power because I must multiply it by itself. So What I do is create a variable to hold the matrix like so
for (i = 0; i < matrixARowSize; i++)
{
for (j = 0; j < matrixAColumnSize; j++)
{
for (k = 0; k < matrixARowSize; k++)
{
sum += matrixA[i][j] * matrixA[i][j];
}
matrixC[i][j] = sum;
sum = 0;
}
}
Then I would have to multiply to itself as much as the user wants to.
Eg:
matrixC[i][j] * matrixC[i][j]*matrixC[i][j] ...// etc
up to whatever power the user wants. I can do that with many If statements yes, but I also need to be able to add them together like so:
matrixC^6 + matrixC^5 + matrixC^4 ...
etc from whatever power the user wants. (Highest is 6).
Any suggestions on how to do this?
You can do this:
int raiseMethod(int val, int pow) {
int temp = val;
for (int i = 1; i < pow; i++) {
temp *= val;
}
return temp;
}
for (int i = 0; i < arrayColummns; i++) {
for (int j = 0; j < arrayRows; j++) {
array[i][j] = raiseMethod(array[i][j], powerToRaise);
}
}
This way, the array will be auto-updated with it's raised value on each position.
I believe you are looking for the Math.pow() method, which raises one number to the power of another, e.g.
sum += (int) Math.power(matrixA[i][j], raiseByPower);
You can do the binary multiplication of the matrix.
This Matrix structure contains everything you need.
#include <stdio.h>
#include <string.h>
const int SIZE = 6;
struct Matrix
{
int m[SIZE][SIZE];
Matrix()
{
memset(m,0,sizeof(m));
}
Matrix( int a[SIZE][SIZE] )
{
for(int i = 0;i<SIZE;++i)for(int j = 0;j<SIZE;++j)
{
m[i][j] = a[i][j];
}
}
Matrix operator * ( const Matrix &a )
{
Matrix ret;
for(int k = 0;k<SIZE;++k) for(int i = 0;i<SIZE;++i) for(int j = 0;j<SIZE;++j)
{
ret.m[i][j] += m[i][k] * a.m[k][j];
}
return ret;
}
Matrix operator ^ ( int P )
{
Matrix ret , a(this->m);
for(int i = 0;i<SIZE;++i)
ret.m[i][i] = 1;
while(P)
{
if( P&1 )
ret = ret * a;
a = a * a;
P >>= 1;
}
return ret;
}
Matrix operator + (const Matrix &a)
{
Matrix ret;
for(int i = 0;i<SIZE;++i) for(int j = 0;j<SIZE;++j)
{
ret.m[i][j] = m[i][j] + a.m[i][j];
}
return ret;
}
};
You can use this structure like the following:
Matrix A, B;
Matrix res = (A^6) + (B^5);
This Power function does log(n) multiplications of matrix.