/* Program to demonstrate gaussian <strong class="highlight">elimination</strong>
on a set of linear simultaneous equations
*/
#include <iostream>
#include <cmath>
#include <vector>
using namespace std;
const double eps = 1.e-15;
/*Preliminary pivoting strategy
Pivoting function
*/
double pivot(vector<vector<double> > &a, vector<double> &b, int i)
{
int n = a.size();
int j=i;
double t=0;
for(int k=i; k<n; k+=1)
{
double aki = fabs(a[k][i]);
if(aki>t)
{
t=aki;
j=k;
}
}
if(j>i)
{
double dummy;
for(int L=0; L<n; L+=1)
{
dummy = a[i][L];
a[i][L]= a[j][L];
a[j][L]= dummy;
}
double temp = b[j];
b[i]=b[j];
b[j]=temp;
}
return a[i][i];
}
/* Forward <strong class="highlight">elimination</strong> */
void triang(vector<vector<double> > &a, vector<double> &b)
{
int n = a.size();
for(int i=0; i<n-1; i+=1)
{
double diag = pivot(a,b,i);
if(fabs(diag)<eps)
{
cout<<"zero det"<<endl;
return;
}
for(int j=i+1; j<n; j+=1)
{
double mult = a[j][i]/diag;
for(int k = i+1; k<n; k+=1)
{
a[j][k]-=mult*a[i][k];
}
b[j]-=mult*b[i];
}
}
}
/*
DOT PRODUCT OF TWO VECTORS
*/
double dotProd(vector<double> &u, vector<double> &v, int k1,int k2)
{
double sum = 0;
for(int i = k1; i <= k2; i += 1)
{
sum += u[i] * v[i];
}
return sum;
}
/*
BACK SUBSTITUTION STEP
*/
void backSubst(vector<vector<double> > &a, vector<double> &b, vector<double> &x)
{
int n = a.size();
for(int i = n-1; i >= 0; i -= 1)
{
x[i] = (b[i] - dotProd(a[i], x, i + 1, n-1))/ a[i][i];
}
}
/*
REFINED GAUSSIAN <strong class="highlight">ELIMINATION</strong> PROCEDURE
*/
void gauss(vector<vector<double> > &a, vector<double> &b, vector<double> &x)
{
triang(a, b);
backSubst(a, b, x);
}
// EXAMPLE MAIN PROGRAM
int main()
{
int n;
cin >> n;
vector<vector<double> > a;
vector<double> x;
vector<double> b;
for (int i = 0; i < n; i++) {
vector<double> temp;
for (int j = 0; j < n; j++) {
int no;
cin >> no;
temp.push_back(no);
}
a.push_back(temp);
b.push_back(0);
x.push_back(0);
}
/*
for (int i = 0; i < n; i++) {
int no;
cin >> no;
b.push_back(no);
x.push_back(0);
}
*/
gauss(a, b, x);
for (size_t i = 0; i < x.size(); i++) {
cout << x[i] << endl;
}
return 0;
}
上述高斯消除算法在NxN矩阵上工作正常。但我需要它来处理NxM矩阵。任何人都可以帮我做吗?我不擅长数学。我在一些网站上得到了这个代码,我坚持不懈。
答案 0 :(得分:17)
答案 1 :(得分:3)
您无法直接将高斯消除应用于NxM问题。如果你有更多的方程式而不是未知数,那么你的问题是过分确定的,你没有解决方案,这意味着你需要使用像最小二乘法这样的方法。假设你有A * x = b,那么你不必拥有x = inv(A)* b(当N = M时),那么你必须做x = inv(A ^ T * A)* A ^ T * b
如果你的方程式较少,那么未知数,那么你的问题是不确定的,你有无穷无尽的解决方案。在这种情况下,您可以随机选择一个(例如将一些未知数设置为任意值),或者您需要使用正则化,这意味着尝试添加一些额外的约束。
答案 2 :(得分:0)
您可以像在此代码段中那样应用梯队缩减
#include <iostream>
#include <algorithm>
#include <vector>
#include <iomanip>
using namespace std;
/*
A rectangular matrix is in echelon form(or row echelon form) if it has the following
three properties :
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of
the row above it.
3. All entries in a column below a leading entry are zeros.
If a matrix in echelon form satisfies the following additional conditions,
then it is in reduced echelon form(or reduced row echelon form) :
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
*/
template <typename C> void print(const C& c) {
for (const auto& e : c) {
cout << setw(10) << right << e;
}
cout << endl;
}
template <typename C> void print2(const C& c) {
for (const auto& e : c) {
print(e);
}
cout << endl;
}
// input matrix consists of rows, which are vectors of double
vector<vector<double>> Gauss::Reduce(const vector<vector<double>>& matrix)
{
if (matrix.size() == 0)
throw string("Empty matrix");
auto A{ matrix };
auto mima = minmax_element(A.begin(), A.end(), [](const vector<double>& a, const vector<double>& b) {return a.size() < b.size(); });
auto mi = mima.first - A.begin(), ma = mima.second - A.begin();
if (A[mi].size() != A[ma].size())
throw string("All rows shall have equal length");
size_t height = A.size();
size_t width = A[0].size();
if (width == 0)
throw string("Only empty rows");
for (size_t row = 0; row != height; row++) {
cout << "processing row " << row << endl;
// Search for maximum below current row in column row and move it to current row; skip this step on the last one
size_t col{ row }, maxRow{ 0 };
// find pivot for current row (partial pivoting)
while (col < width)
{
maxRow = distance(A.begin(), max_element(A.begin() + row, A.end(), [col](const vector<double>& rowVectorA, const vector<double>& rowVectorB) {return abs(rowVectorA[col]) < abs(rowVectorB[col]); }));
if (A[maxRow][col] != 0) // nonzero in this row and column or below found
break;
++col;
}
if (col == width) // e.g. in current row and below all entries are zero
break;
if (row != maxRow)
{
swap(A[row], A[maxRow]);
cout << "swapped " << row << " and " << maxRow;
}
cout << " => leading entry in column " << col << endl;
print2(A);
// here col >= row holds; col is the column of the leading entry e.g. first nonzero column in current row
// moreover, all entries to the left and below are zeroed
if (row+1 < height)
cout << "processing column " << col << endl;
// Make in all rows below this one 0 in current column
for (size_t rowBelow = row + 1; rowBelow < height; rowBelow++) {
// subtract product of current row by factor
double factor = A[rowBelow][col] / A[row][col];
cout << "processing row " << rowBelow << " below the current; factor is " << factor << endl;
if (factor == 0)
continue;
for (size_t colRight{ col }; colRight < width; colRight++)
{
auto d = A[rowBelow][colRight] - factor * A[row][colRight];
A[rowBelow][colRight] = abs(d) < DBL_EPSILON ? 0 : d;
}
print(A[rowBelow]);
}
}
// the matrix A is in echelon form now
cout << "matrix in echelon form" << endl;
print2(A);
// reduced echelon form follows (backward phase)
size_t row(height-1);
auto findPivot = [&row, A] () -> size_t {
do
{
auto pos = find_if(A[row].begin(), A[row].end(), [](double d) {return d != 0; });
if (pos != A[row].end())
return pos - A[row].begin();
} while (row-- > 0);
return A[0].size();
};
do
{
auto col = findPivot();
if (col == width)
break;
cout << "processing row " << row << endl;
if (A[row][col] != 1)
{
//scale row row to make element at [row][col] equal one
auto f = 1 / A[row][col];
transform(A[row].begin()+col, A[row].end(), A[row].begin()+col, [f](double d) {return d * f; });
}
auto rowAbove{ row};
while (rowAbove > 0)
{
rowAbove--;
double factor = A[rowAbove][col];
if (abs(factor) > 0)
{
for (auto colAbove{ 0 }; colAbove < width; colAbove++)
{
auto d = A[rowAbove][colAbove] - factor * A[row][colAbove];
A[rowAbove][colAbove] = abs(d) < DBL_EPSILON ? 0 : d;
}
cout << "transformed row " << rowAbove << endl;
print(A[rowAbove]);
}
}
} while (row-- > 0);
return A;
}