Math utility functions. More...

Functions
double	golden_section (double RHS(double x, void context), double a, double b, double tol, int maxIts, void context)
	Golden section minimization.
ml_matrix	rk0 (ml_matrix RHS(ml_matrix x, double t, void context), ml_matrix &x, double h, double t, void context)
	0th order Runge-Kutta integration algorithm with time as a RHS input
ml_matrix	rk4 (ml_matrix RHS(ml_matrix x, double t), ml_matrix &x, double h, double t)
	4th order Runge-Kutta integration algorithm with time as a RHS input
ml_matrix	rk4 (ml_matrix RHS(ml_matrix x, double t, void context), ml_matrix &x, double h, double t, void context)
	4th order Runge-Kutta integration algorithm with time as a RHS input and a context pointer
ml_matrix	rk4 (ml_matrix RHS(ml_matrix &x, ml_matrix &a), ml_matrix &x, double h, ml_matrix a)
	4th order Runge-Kutta integration algorithm with a matrix input to the RHS
double	pss_rem (double x, double y)
	Remainder function.
double	fix (double x)
	Round to 0.
double	r2p5 (double x)
	Round to nearest 0.5.
int	sct_sign (double x)
	Determine sign of number and return unitary value.
double	sct_max (double a, double b)
	Find maximum of two numbers.
double	unwrap (double angle)
	Unwrap radian values to (-pi,pi) interval.
ml_matrix	intersect_line_ellipsoid (const ml_matrix &p, const ml_matrix &u, const ml_matrix &e)
	Find the intersection between a line and an ellipsoid.
ml_matrix	p_gauss (int nMax, int mMax, double theta, ml_matrix &dp)
	Computes the Gaussian form of the Legendre functions and the first derivatives.
void	s_c_harm (const ml_matrix &a, int n, ml_matrix &s, ml_matrix &c)
	Generate a series of sine and cosine harmonics.
void	c_to_d_zoh (const ml_matrix &a, const ml_matrix &b, double dT, ml_matrix &aD, ml_matrix &bD)
	Compute a discrete-time linear system from a continuous system.
ml_matrix	expm (const ml_matrix &a)
	Compute the matrix exponential (of a square matrix only) using a power series expansion.
double	WrapPhase (double angle)
	Wrap a phase angle to keep its value between -pi and +pi.
ml_matrix	WrapPhase (ml_matrix angle)
	Wrap a vector of phase angles to keep their values between -pi and +pi.
ml_matrix	UnwrapPhase (ml_matrix theta)
	Unwrap a vector of angular values so they change continuously instead of wrapping.
ml_matrix	diff (ml_matrix a)
	Compute the difference between successive elements of a 1-D vector.
ml_matrix	delt_circ (ml_matrix x, ml_matrix y)
	Compute a delta in circular coordinates.
ml_matrix	circle_fit (ml_matrix x, ml_matrix y)
	Compute a delta in circular coordinates.
double	factorial (double n)
	Compute a factorial.
double	interp_2d (ml_matrix a, ml_matrix x, ml_matrix y, double xK, double yK)
	2D interpolation
double	newton_raphson (double f(double x, void context), double df(double x, void context), double x0, double tol, int nMax, void *context)
	1 dimensional root finder
double	secant (double RHS(double x, void context), double x0, double x00, double tol, int nMax, void context)
	1 dimensional root finder
ml_matrix	average_unit_vector (ml_matrix u1, ml_matrix u2)
	Average unit vector.
ml_matrix	pinv (ml_matrix a)
	Pseudo inverse.
ml_matrix	downhill_simplex (double RHS(const ml_matrix &x, void context), ml_matrix &x, ml_matrix &options, void context)
	Downhill simplex.
ml_matrix	orthogonalize (ml_matrix a)
	Orthogonalize.
ml_matrix	logistic (ml_matrix x)
	Logistic function.
double	logistic (double x)
	Logistic function.
double	interp_extrap_1d (ml_matrix a, ml_matrix x, double xK)
	1D interpolation and extrapolation
double	gaussian_multivariate_density (ml_matrix y, ml_matrix y_bar, ml_matrix P)
	Gaussian Mulivariate Normal Distribution function.
ml_matrix	roots_second_order (double a, double b, double c)
	Roots second order.
ml_matrix	roots_second_order (const ml_matrix &p)
	Roots second order.
double	trig_reduction (double a, double b, double c)
	Solve for beta: asin(beta)+bcos(beta) = c.

Detailed Description

Function Documentation

double golden_section	(	double	RHSdouble x, void *context,
		double	a,
		double	b,
		double	tol,
		int	maxIts,
		void *	context
	)

Parameters:

RHS	Right-hand side function definition
a	Left range
b	Right range
tol	Tolerance
maxIts	Maximum number of interations
*context	to data from RSH

Returns:: minimum value of x

ml_matrix rk0	(	ml_matrix	RHSml_matrix x, double t, void *context,
		ml_matrix &	x,
		double	h,
		double	t,
		void *	context
	)

0th order Runge-Kutta integration algorithm with time as a RHS input

RHS actually does the propagation

Parameters:

RHS	Right-hand side function definition
x	Current state
h	Integration time step
t	Current time
*context	Pointer to data to be passed to RHS

Returns:: Derivative of x

ml_matrix rk4	(	ml_matrix	RHSml_matrix x, double t,
		ml_matrix &	x,
		double	h,
		double	t
	)

4th order Runge-Kutta integration algorithm with time as a RHS input

Parameters:

RHS	Right-hand side function definition
x	Current state
h	Integration time step
t	Current time

Returns:: Derivative of x

double pss_rem	(	double	x,
		double	y
	)

Remainder function.

Parameters:

x	Numerator
y	Denominator

Returns:: Remainder from the division of y into x

double fix ( double x )

Round to 0.

Parameters:

x	Number to be rounded

Returns:: Number rounded to 0

double r2p5 ( double x )

Round to nearest 0.5.

Parameters:

x	Number to be rounded

Returns:: Number rounded to the nearest "point 5"

int sct_sign ( double x )

Determine sign of number and return unitary value.

Parameters:

x Number

Returns:: Unitary value, i.e. 1 or -1

double sct_max	(	double	a,
		double	b
	)

Find maximum of two numbers.

Parameters:

a	First number
b	Second number

Returns:: Maximum of a and b

double unwrap ( double angle )

Unwrap radian values to (-pi,pi) interval.

Parameters:

angle Angle in radians

Returns:: Unwrapped angle

References PI, and TWO_PI.

ml_matrix intersect_line_ellipsoid	(	const ml_matrix &	p,
		const ml_matrix &	u,
		const ml_matrix &	e
	)

Always returns the closest point. The line is defined by a point and unit vectors.

Parameters:

p	(3,1) Point
u	(3,1) Unit vector
e	(3,1) (a,b,c) semi-axes of ellipsoid

Returns:: Intersection point

ml_matrix p_gauss	(	int	nMax,
		int	mMax,
		double	theta,
		ml_matrix &	dp
	)

Computes the Gaussian form of the Legendre functions and the first derivatives.

Because there is no zero indexing in MATLAB, the P's are offset in the p matrix as follows:

n,m p(n+1,m+1) = P

MATLAB Form: [p, dP] = PGauss( nMax, mMax, theta )

------------------------------------------------------------------------ References: Wertz, J., Spacecraft Attitude Determination and Control, Kluwer, 1976, pp. 781.

Parameters:

nMax	max value of first index
mMax	max value of second index
theta	input value (usually an angle in rads)
dp	Matrix for derivative outputs dp(n,m) / d(theta)

Returns:: p Gauss functions, P(n,m)

void s_c_harm	(	const ml_matrix &	a,
		int	n,
		ml_matrix &	s,
		ml_matrix &	c
	)

Generate a series of sine and cosine harmonics.

Parameters:

a	Column vector (m,1) argument, in radians
n	Number of harmonics
s	Vector of sine harmonics output
c	Vector of cosine harmonics output

void c_to_d_zoh	(	const ml_matrix &	a,
		const ml_matrix &	b,
		double	dT,
		ml_matrix &	f,
		ml_matrix &	g
	)

Compute a discrete-time linear system from a continuous system.

Parameters:

a	Continuous matrix
b	Continuous matrix
dT	Time step (sec)
f	Discrete matrix
g	Discrete matrix

ml_matrix expm ( const ml_matrix & a )

Parameters:

a	Square matrix

Returns:: b Matrix exponential, b = exp(a)

double WrapPhase ( double angle )

*sctlib

Parameters:

angle Phase angle [rad]

Returns:: Phase angle wrapped between -PI and PI [rad]

References PI, and TWO_PI.

ml_matrix WrapPhase ( ml_matrix angle )

*sctlib

Parameters:

angle Vector of phase angles with no limits. [rad]

Returns:: wrapped Vector of phase angles wrapped between -PI and +PI. [rad]

References PI, and TWO_PI.

ml_matrix UnwrapPhase ( ml_matrix angle )

*sctlib

Parameters:

angle Vector of angular values. [rad]

Returns:: unwrapped New vector with jumps larger than 2*PI in magnitude removed so that it changes "continuously". [rad]

References PI, and TWO_PI.

ml_matrix diff ( ml_matrix a )

Compute the difference between successive elements of a 1-D vector.

Parameters:

a	1-dimensional matrix, length N

Returns:: b 1-dimensional matrix, ith element is a(i+1)-a(i), length N-1

ml_matrix delt_circ	(	ml_matrix	x,
		ml_matrix	y
	)

Compute a delta in circular coordinates.

It does not check dimensions.

Parameters:

x	2-dimensional matrix
y	2-dimensional matrix

Returns:: z 2-dimensional matrix

References PI.

ml_matrix circle_fit	(	ml_matrix	x,
		ml_matrix	y
	)

Compute a delta in circular coordinates.

Parameters:

x	column matrix
y	column matrix

Returns:: s column matrix [a;b;r]

ml_matrix pinv ( ml_matrix a )

Pseudo inverse.

Parameters:

a matrix

Returns:: orthogonal matrix

ml_matrix orthogonalize ( ml_matrix a )

Orthogonalize.

Orthogonalize a 3x3 matrix

Parameters:

a matrix

Returns:: orthogonal matrix

ml_matrix logistic ( ml_matrix x )

Parameters:

x input

Returns:: 1/(1+exp(-x))

double logistic ( double x )

Parameters:

x input

Returns:: 1/(1+exp(-x))

double interp_extrap_1d	(	ml_matrix	a,
		ml_matrix	x,
		double	xK
	)

1D interpolation and extrapolation

Parameters:

a	array of values
x	corresponding x value
xK	value of interest

Returns:: value

double gaussian_multivariate_density	(	ml_matrix	y,
		ml_matrix	y_bar,
		ml_matrix	P
	)

Gaussian Mulivariate Normal Distribution function.

Parameters:

y	array of parameters
y_bar	corresponding means
P	covariance matrix

Returns:: likelihood

ml_matrix roots_second_order	(	double	a,
		double	b,
		double	c
	)

Roots second order.

Parameters:

a	Coefficient
b	Coefficient
c	Coefficient

Returns:: roots

ml_matrix roots_second_order ( const ml_matrix & p )

Roots second order.

Parameters:

p	array of coefficients

Returns:: roots

double trig_reduction	(	double	a,
		double	b,
		double	c
	)

Parameters:

a	Coefficient
b	Coefficient
c	RHS

Returns:: roots

sc_math.h File Reference

Functions

Detailed Description

Function Documentation