So far I have this code for an LU decomposition. It takes in an input array and it returns the lower and upper triangular matrix.
void LUFactorization ( int d, const double*S, double*L, double*U )
{
for(int k = 0; k < d; ++k){
if (
for(int j = k; j < d; ++j){
double sum = 0.;
for(int p = 0; p < k; ++p) {
sum+=L[k*d+p]*L[p*d+j];
cout << L[k*d+p] << endl;
}
sum = S[k*d+j] - sum;
L[k*d+j]=sum;
U[k*d+j]=sum;
}
for(int i = k + 1; i < d; ++i){
double sum=0.;
for(int p = 0; p < k; ++p) sum+=L[i*d+p]*L[p*d+k];
L[i*d+k]=(S[i*d+k]-sum)/L[k*d+k];
}
}
for(int k = 0; k < d; ++k){
for(int j = k; j < d; ++j){
if (k < j) L[k*d+j]=0;
else if (k == j) L[k*d+j]=1;
}
}
}
Is there some way I can adapt this to perform row exchanges? If not, is there some other algorithm I could be directed towards?
Thanks
The usual approach for LU decompositions with pivoting is to store a permutation array P which is initialized as the identity permutation (P={0,1,2,...,d - 1}) and then swapping entries in P instead of swapping rows in S
If you have this permutation array, every access to S must use P[i] instead of i for the row number.
Note that P itself is part of the output, since it represents the permutation matrix such that
P*A = L*U, so if you want to use it to solve systems of linear equations, you'll have to apply P on the right-hand side before applying backward and forward substitution
Related
Given an m x n integer grid, return the size (i.e., the side length k) of the largest magic square that can be found within this grid.
The question can be found here on leetcode
I first wanted to see if a naive brute force approach would pass, so I came up with the following algorithm
Iterate through all values of k (from min(rows,cols) of the matrix to 1)
For each of the k values, check if it's possible to create a magic of square of dimensions kxk by checking all possible sub matrices and
return k if it's possible. This would be O(rows*cols*k^2)
So that would make the overall complexity O(k^3*rows*cols). (Please correct me if I am wrong)
I have attached my code in C++ below
class Solution {
public:
int largestMagicSquare(vector<vector<int>>& grid) {
int rows = grid.size(),cols = grid[0].size();
for(int k = min(rows,cols); k >= 2; k--){ // iterate over all values of k
for(int i = 0; i < rows-k+1; i++){
for(int j = 0; j < cols-k+1; j++){
int startX = i, startY = j, endX = i+k-1, endY = j+k-1;
int diagSum = 0, antiDiagSum = 0;
bool valid = true;
// calculate sum of diag
for(int count = 0; count < k; count++){
diagSum += grid[startX][startY];
startX++,startY++;
}
// this is the sum that must be same across all rows, cols, diag and antidiag
int sum = diagSum;
// calculate sum of antidiag
for(int count = 0; count < k; count++){
antiDiagSum += grid[endX][endY];
endX--,endY--;
}
if(antiDiagSum != sum) continue;
// calculate sum across cols
for(int r = i; r <=i+k-1; r++){
int colSum = 0;
for(int c = j; c <= j+k-1; c++){
colSum += grid[r][c];
}
if(colSum != sum){
valid = false;
break;
}
}
if(!valid) continue;
// calculate sum across rows
for(int c = j; c <= j+k-1; c++){
int rowSum = 0;
for(int r = i; r <= i+k-1; r++){
rowSum += grid[r][c];
}
if(rowSum != sum){
valid = false;
break;
}
}
if(!valid) continue;
return k;
}
}
}
return 1;
}
};
I thought I would optimize the solution once this works (Maybe binary search over the values of k). However, my code is failing for a really large test case for a matrix of dimension 50x50 after passing 74/80 test cases on Leetcode.
I tried to find out the source(s) that could be causing it to fail, but I am not really sure where the error is.
Any help would be appreciated. Thanks!
Please do let me know if further clarification about the code is needed
The calculation of antiDiagSum is wrong: it actually sums the values on the same diagonal as diagSum, just in reverse order. To traverse the opposite diagonal, you need to increment the Y coordinate and decrement the X coordinate (or vice versa), but your code decrements both of them.
It is probably easiest if you fix this by calculating both diagonal sums in the same loop:
for(int count = 0; count < k; count++){
diagSum += grid[startX][startY];
antiDiagSum += grid[endX][startY];
startX++, startY++, endX--;
}
I'm trying to implement a quick program to solve a system of linear equations. The program reads the input from a file and then writes the upper-triangular system and solutions to a file. It is working with no pivoting, but when I try to implement the pivoting it produces incorrect results.
As example input, here is the following system of equations:
w+2x-3y+4z=12
2w+2x-2y+3z=10
x+y=-1
w-x+y-2z=-4
I expect the results to be w=1, x=0, y=-1 and z=2. When I don't pivot, I get this answer (with some rounding error on x). When I add in the pivoting, I get the same numbers but in the wrong order: w=2,x=1,y=-1 and z=0.
What do I need to do to get these in the correct order? Am I missing a step somewhere? I need to do column swapping instead of rows because I need to adapt this to a parallel algorithm later that requires that. Here is the code that does the elimination and back substitution:
void gaussian_elimination(double** A, double* b, double* x, int n)
{
int maxIndex;
double temp;
int i;
for (int k = 0; k < n; k++)
{
i = k;
for (int j = k+1; j < n; j++)
{
if (abs(A[k][j]) > abs(A[k][i]))
{
i = j;
}
}
if (i != k)
{
for (int j = 0; j < n; j++)
{
temp = A[j][k];
A[j][k] = A[j][i];
A[j][i] = temp;
}
}
for (int j = k + 1; j < n; j++)
{
A[k][j] = A[k][j] / A[k][k];
}
b[k] = b[k] / A[k][k];
A[k][k] = 1;
for (i = k + 1; i < n; i++)
{
for (int j = k + 1; j < n; j++)
{
A[i][j] = A[i][j] - A[i][k] * A[k][j];
}
b[i] = b[i] - A[i][k] * b[k];
A[i][k] = 0;
}
}
}
void back_substitution(double**U, double*x, double*y, int n)
{
for (int k = n - 1; k >= 0; k--)
{
x[k] = y[k];
for (int i = k - 1; i >= 0; i--)
{
y[i] = y[i] - x[k]*U[i][k];
}
}
}
I believe what you implemented is actually complete pivoting.
With complete pivoting, you must keep track of the permutation of columns, and apply the same permutation to your answer.
You can do this with an array {0, 1, ..., n}, where you swap the i'th and k'th values in the second loop. Then, rearange the solution using this array.
If what you were trying to do is partial pivoting, you need to look for the maximum in the respective row, and swap the rows and the values of 'b' accordingly.
I'm trying to design a program that creates a matrix using vectors of vectors of integers, and then multiplyies it with another matrix. I know how to multiply matrices on paper, but when I try to implement it in my program, I'm not getting it to work. I know that both matrices are entered correctly and are passed correctly as I have the the output of those functions so that I can debug. The program works incorrectly when I try to multiply them. The answer and the number of elements are not right. I know I'm missing something but can't figure out what.
Matrix Matrix::operator*(Matrix m){
vector<int> mRow = m.getRow(0);
vector<int> mCol = m.getCol(0);
vector<int> newElem;
int product = 0;
//adds the contents of the 2nd matrix to the 2d vector
vector< vector<int> > m2(mRow.size(), vector<int>(mCol.size()));
for (int i = 0; i < mRow.size(); i++){
mRow.clear();
mRow = m.getRow(i);
for (int j = 0; j < mCol.size(); j++){
m2[j][i] = mRow[j];
}
}
//Multiplies the matrices using the 2d matrix**THIS IS WHERE IT GOES WRONG**
for (int i = 0; i < row; i++){
for (int j = 0; j < column; j++){
product += matrix[i][j]*m2[j][i];
}
newElem.insert(newElem.begin()+i,product);
product = 0;
}
//displays the products so that i can see if its working
for (int i = 0; i < newElem.size(); i++){
cout << " "<<newElem[i]<<endl;
}
//adds the new product vector to a new Matrix object and returns it
Matrix newM(row, mCol.size());
vector<int> temp;
for (int i = 0; i < row; i++){
for (int j = 0; j < mCol.size(); j++){
temp.insert(temp.begin()+j, newElem[0]);
newElem.erase(newElem.begin());
}
newM.setRow(temp,i);
temp.clear();
}
return newM;
}
Although I don't know whether this helps, I'm using this site as a reference for multiplying 2 matrices together.
Your matrix representation has nothing to do with your mistake. You need to have more nested iterations. Think of a result matrix and iterate through that to calculate it's every element. In a pseudocode:
for i in result column
for j in result row
res[i, j] = multiply(m1, m2, i, j)
where multiply function is the nested loop, something like this:
multiply(m1, m2, i, j)
{
val = 0;
for k in row
val += m1[i, k] * m2[k, j]
return val
}
Here is an implementation of the outer loops. Mind you, there are no error checking in the code.
vector<vector<int> > ml;
vector<vector<int> > mr;
// fill in ml and mr
...
// result matrix
vector<vector<int> > res;
// allocate the result matrix
res.resize(ml.size());
for( it = res.begin(); it != res.end(); ++it)
it->resize(ml[0].size());
// loop through the result matrix and fill it in
for( int i = 0; i < res.size(); ++i)
for( int j = 0; j < res[0].size(); ++j)
res[i][j] = multiply(ml, mr, i, j);
Leaving a proper implementation of multiply() function to you.
i want to potentiate a Matrix but i dont workings how it should work.
m ist the Matrix i want to potentiate
long double pro[100][100]; // product after each step
long double res[100][100]; // the Matrix with the exponent n
for (int n = 1; n < nVal; n++) // exponent
{
for (int i = 0; i < mVal; i++) // row
{
for (int j = 0; j < mVal; j++) // col
{
res[i][j] = 0;
for (int k = 0; k < mVal; k++) // inner
{
res[i][j] += pro[i][k] * m[k][j]; // multiply the product with the default matrix
}
}
}
}
// array Output - working
for (int i = 0; i<mVal; i++)
{
for (int j = 0; j<mVal; j++)
cout << res[i][j] << "\t";
cout << endl;
}
in the output i see some crazy numbers and i dont know why :(
Can anyone help me?
You should
initialise the pro matrix to the identity at the beginning of loop on n
copy the res matrix into the pro matrix the end of each loop on n.
In pseudo code
pro = Identity matrix
for (int n = 1; n < nVal; n++) {
res = pro * m // using two loops
pro = res
}
result is in pro.
Note that there are much faster way to compute powers: http://en.wikipedia.org/wiki/Exponentiation_by_squaring
As Willll said you shouldn't forget to initialize.
Another suggestion would be to erase the exponent loop and just use the pow() function from math library. It´ll make it more simple and easier to visualize.
So what I am trying to do is multiply one 2d vector by another 2d vector.
I come from Java,Python and C# so I am pretty much learning C++ as I go along.
I have the code down to generate the vector and display the vector but I can't seem to finish the multiplication part.
v1 is another matrix that is already generated.
vector<vector<int> > v2 = getVector();
int n1 = v1[0].size();
int n2 = v2.size();
vector<int> a1(n2, 0);
vector<vector<int> > ans(n1, a1);
for (int i = 0; i < n1; i++) {
for (int j = 0; j < n2; j++) {
for (int k = 0; k < 10; k++) {
// same as z[i][j] = z[i][j] + x[i][k] * y[k][j];
ans[i][j] += v1[i][k] * v2[k][j];
}
}
}
displayVector(ans);
My guess for where I am going wrong is in the inner-most loop. I can't figure out what to actually put in place of that 10 I have there now.
When you multiply matrices, the number of columns of the matrix on the left side must equal the number of rows of the matrix on the right side. You need to check that that is true, and use that common number for your size of the k variable:
int nCommon = v1.size();
assert(v2[0].size() == nCommon);
for (int i = 0; i < n1; i++) {
for (int j = 0; j < n2; j++) {
for (int k = 0; k < nCommon ; k++) {
ans[i][j] += v1[i][k] * v2[k][j];
}
}
}
For you inner loop, you should do something like this
ans[i][j] = 0;
for (int k = 0; k < n2; k++) {
ans[i][j] += v1[i][k] * v2[k][j];
}
I don't know where the 10 comes from.