Double-Valued Vectors ( vector )

Definition

An instance of data type vector is a vector of variables of type double.

#include < LEDA/numbers/vector.h >

Creation

vector	v	creates an instance v of type vector; v is initialized to the zero-dimensional vector.
vector	v(int d)	creates an instance v of type vector; v is initialized to the zero vector of dimension d.
vector	v(double a, double b)	creates an instance v of type vector; v is initialized to the two-dimensional vector (a, b).
vector	v(double a, double b, double c)
		creates an instance v of type vector; v is initialized to the three-dimensional vector (a, b, c).
vector	v(const vector& w, int prec)
		creates an instance v of type vector; v is initialized to a copy of w. The second argument is for compatibility with rat_vector.

Operations

int	v.dim()	returns the dimension of v.
double&	v[int i]	returns i-th component of v. Precondition 0 < = i < = v.dim()-1.
double	v.hcoord(int i)	for compatibility with rat_vector.
double	v.coord(int i)	for compatibility with rat_vector.
double	v.sqr_length()	returns the square of the Euclidean length of v.
double	v.length()	returns the Euclidean length of v.
vector	v.norm()	returns v normalized.
double	v.angle(const vector& w)	returns the angle between v and w.
vector	v.rotate90(int i=1)	returns v by an angle of i x 90 degrees. If i > 0 the rotation is counter-clockwise otherwise it is clockwise. Precondition v.dim() = 2
vector	v.rotate(double a)	returns the v rotated counter-clockwise by an angle of a (in radian). Precondition v.dim() = 2
vector&	v += const vector& v1	Addition and assign. Precondition v.dim() = v1.dim().
vector&	v -= const vector& v1	Subtraction and assign. Precondition v.dim() = v1.dim().
vector	v + const vector& v1	Addition. Precondition v.dim() = v1.dim().
vector	v - const vector& v1	Subtraction. Precondition v.dim() = v1.dim().
double	v * const vector& v1	Scalar multiplication. Precondition v.dim() = v1.dim().
vector	v * double r	Componentwise multiplication with double r.
vector&	v *= double r	multiplies all coordinates by r.
vector	v / double r	Componentwise division which double r.
bool	v == const vector& w	Test for equality.
bool	v != const vector& w	Test for inequality.
void	v.print(ostream& O)	prints v componentwise to ostream O.
void	v.print()	prints v to cout.
void	v.read(istream& I)	reads d = v.dim() numbers from input stream I and writes them into v[0]...v[d - 1].
void	v.read()	reads v from cin.
ostream&	ostream& O « const vector& v
		writes v componentwise to the output stream O.
istream&	istream& I » vector& v	reads v componentwise from the input stream I.

Additional Operations for vectors in two and three-dimensional space

double	v.xcoord()	returns the zero-th cartesian coordinate of v.
double	v.ycoord()	returns the first cartesian coordinate of v.
double	v.zcoord()	returns the second cartesian coordinate of v.
int	compare_by_angle(const vector& v1, const vector& v2)
		For a non-zero vector v let $\alpha$(v) be the angle by which the positive x-axis has to be turned counter-clockwise until it aligns with v. The function compares the angles defined by v1 and v2, respectively. The zero-vector precedes all non-zero vectors in the angle-order.
vector	cross_product(const vector& v1, const vector& v2)
		returns the cross product of the three-dimensional vectors v1 and v2.

Implementation

Vectors are implemented by arrays of real numbers. All operations on a vector v take time O(v.dim()), except for dim and [ ] which take constant time. The space requirement is O(v.dim()).

Be aware that the operations on vectors and matrices incur rounding errors 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 necessarily true that the test b == M * x evaluates to true. The types integer_vector and integer_matrix provide exact linear algebra.