Real-Valued Matrices (matrix)

Definition

An instance of the data type matrix is a matrix of variables of type double.

Creation

matrix	M(int n=0, int m=0);	creates an instance M of type matrix, M is initialized to the n x m - zero matrix.
matrix	M(int n, int m, double* D);
		creates the nx m matrix M with M(i,j) = D[i*m +j] for 0 <= i <= n-1 and 0 <= j <= m-1. Precondition: D points to an array of at least n*m numbers of type double.

Operations

int	M.dim1()	returns n, the number of rows of M.
int	M.dim2()	returns m, the number of columns of M.
vector&	M.row(int i)	returns the i-th row of M (an m-vector). Precondition: 0 <= i <= n-1.
vector	M.col(int i)	returns the i-th column of M (an n-vector). Precondition: 0 <= i <= m-1.
matrix	M.trans()	returns M^T (mx n - matrix).
matrix	M.inv()	returns the inverse matrix of M. Precondition: M.det() != 0.
double	M.det()	returns the determinant of M. Precondition: M is quadratic.
vector	M.solve(vector b)	returns vector x with M* x = b. Precondition: M.dim1() == M.dim2() = =b.dim() and M.det() !=q 0.
double&	M(int i, int j)	returns M_i,j. Precondition: 0<= i<= n-1 and 0<= j<= m-1.
matrix	M + M1	Addition. Precondition: M.dim1() == M1.dim1() and M.dim2() == M1.dim2().
matrix	M - M1	Subtraction. Precondition: M.dim1() == M1.dim1() and M.dim2() == M1.dim2().
matrix	M * M1	Multiplication. Precondition: M.dim2() == M1.dim1().
vector	M * vector vec	Multiplication with vector. Precondition: M.dim2() == vec.dim().
matrix	M * double x	Multiplication with double x.
void	M.print(ostream& O)	prints M row by row to ostream O.
void	M.print()	prints M cout.
void	M.read(istream& I)	reads M.dim1() x M.dim2() numbers from input stream I and writes them row by row into matrix M.
void	M.read()	prints M from cin.
ostream&	ostream& O << M	writes matrix M row by row to the output stream O.
istream&	istream& I >> matrix& M	reads a matrix row by row from the input stream I and assigns it to M.

Implementation

Data type matrix is implemented by two-dimensional arrays of double numbers. Operations det, solve, and inv take time O(n^3), dim1, dim2, row, and col take constant time, all other operations take time O(nm). The space requirement is O(nm).

Be aware that the operations on vectors and matrices incur rounding error and hence are not completely reliable. For example, if M is a matrix, b is a vector, and x is computed by x = M.solve(b) it is not necessarly true that the test b == M * b evaluates to true. The types integer_vector and integer_matrix provide exact linear algebra.