sc_math.h File Reference

Math utility functions. More...

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Functions
double	golden_section (double RHS(double x, void *context), double a, double b, double tol, int maxIts, void *context)
	Golden section minimization. More...
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 More...
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 More...
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. More...
double	fix (double x)
	Round to 0. More...
double	r2p5 (double x)
	Round to nearest 0.5. More...
int	sct_sign (double x)
	Determine sign of number and return unitary value. More...
double	sct_max (double a, double b)
	Find maximum of two numbers. More...
double	unwrap (double angle)
	Unwrap radian values to (-pi,pi) interval. More...
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. More...
ml_matrix	intersect_line_sphere (const ml_matrix &p, const ml_matrix &u, double r)
	Find the intersection between a line and an sphere. More...
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. More...
void	s_c_harm (const ml_matrix &a, int n, ml_matrix &s, ml_matrix &c)
	Generate a series of sine and cosine harmonics. More...
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. More...
ml_matrix	expm (const ml_matrix &a)
	Compute the matrix exponential (of a square matrix only) using a power series expansion. More...
double	WrapPhase (double angle)
	Wrap a phase angle to keep its value between -pi and +pi. More...
ml_matrix	WrapPhase (ml_matrix angle)
	Wrap a vector of phase angles to keep their values between -pi and +pi. More...
ml_matrix	UnwrapPhase (ml_matrix theta)
	Unwrap a vector of angular values so they change continuously instead of wrapping. More...
ml_matrix	diff (ml_matrix a)
	Compute the difference between successive elements of a 1-D vector. More...
ml_matrix	delt_circ (ml_matrix x, ml_matrix y)
	Compute a delta in circular coordinates. More...
ml_matrix	circle_fit (ml_matrix x, ml_matrix y)
	Compute a delta in circular coordinates. More...
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. More...
ml_matrix	downhill_simplex (double RHS(const ml_matrix &x, void *context), ml_matrix &x, ml_matrix &options, void *context)
	Downhill simplex. More...
ml_matrix	orthogonalize (ml_matrix a)
	Orthogonalize. More...
ml_matrix	logistic (ml_matrix x)
	Logistic function. More...
double	logistic (double x)
	Logistic function. More...
double	interp_extrap_1d (ml_matrix a, ml_matrix x, double xK)
	1D interpolation and extrapolation More...
double	gaussian_multivariate_density (ml_matrix y, ml_matrix y_bar, ml_matrix P)
	Gaussian Mulivariate Normal Distribution function. More...
ml_matrix	roots_second_order (double a, double b, double c)
	Roots second order. More...
ml_matrix	roots_second_order (const ml_matrix &p)
	Roots second order. More...
ml_matrix	skew_symmetric (const ml_matrix &p)
	Generate a skew symmetric matrix. More...
double	trig_reduction (double a, double b, double c)
	Solve for beta: a*sin(beta)+b*cos(beta) = c. More...
ml_matrix	m_k_3_by_3 (const ml_matrix &m, int k)
	Returns a 3x3 matrix from a 3x3*n matrix. More...
int	intersect_line_sphere_true (const ml_matrix &p1, const ml_matrix &p2, double r)
	Find if a line intersects a sphere. More...
ml_matrix	append_diagonal (const ml_matrix &a, const ml_matrix &b)
	Append a diagonal matrix to another matrix. More...
double	trapezoidal_integration (double Integrand(double x, void *context), double a, double b, int n, void *context)
	Trapezoidal integration. More...
double	two_dimensional_integration (ml_matrix Integrand(const ml_matrix &x, const ml_matrix &y, void *context), double a, double b, double c, double d, int nX, int nY, void *context)
	Two dimensional integration. More...

Detailed Description

Math utility functions.

Function Documentation

golden_section()

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

Golden section minimization.

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

rk0()

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

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

rk4()

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

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

pss_rem()

double pss_rem	(	double	x,
		double	y
	)

Remainder function.

Remainder function.

Parameters

x	Numerator
y	Denominator

Returns

Remainder from the division of y into x

Referenced by delt_circ(), and r2p5().

fix()

double fix

(

double

x

)

Round to 0.

Round to 0.

Parameters

x	Number to be rounded

Returns

Number rounded to 0

Referenced by jd_to_date().

r2p5()

double r2p5

(

double

x

)

Round to nearest 0.5.

Round to nearest 0.5.

Parameters

x	Number to be rounded

Returns

Number rounded to the nearest "point 5"

References pss_rem().

Referenced by jd_to_date(), and mean_sidereal_day().

sct_sign()

int sct_sign

(

double

x

)

Determine sign of number and return unitary value.

Determine sign of number and return unitary value.

Parameters

x

Number

Returns

Unitary value, i.e. 1 or -1

Referenced by ImpulsiveManeuver(), InPlane(), OutOfPlane(), and Target().

sct_max()

double sct_max	(	double	a,
		double	b
	)

Find maximum of two numbers.

Find maximum of two numbers.

Parameters

a	First number
b	Second number

Returns

Maximum of a and b

unwrap()

double unwrap

(

double

angle

)

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

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

Parameters

angle

Angle in radians

Returns

Unwrapped angle

References PI, and TWO_PI.

Referenced by PID3Axis::Update().

intersect_line_ellipsoid()

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.

Always returns the closest point. The line is defined by a point and unit vectors. Returns empty if there is no intersection

Parameters

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

Returns

Intersection point

Referenced by intersect_line_sphere().

intersect_line_sphere()

ml_matrix intersect_line_sphere	(	const ml_matrix &	p,
		const ml_matrix &	u,
		double	r
	)

Find the intersection between a line and an sphere.

Find the intersection between a line and an sphere.

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
r	(1,1) Radius of sphere

Returns

Intersection point

References intersect_line_ellipsoid().

p_gauss()

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.

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)

Referenced by mag_field().

s_c_harm()

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

Generate a series of sine and cosine harmonics.

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

Referenced by mag_field().

c_to_d_zoh()

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.

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

References expm().

Referenced by FFEccLinOrb(), LPCircular(), LPEccentricGVE(), and MonitoringAlg().

expm()

ml_matrix expm

(

const ml_matrix &

a

)

Compute the matrix exponential (of a square matrix only) using a power series expansion.

Parameters

a	Square matrix

Returns

b Matrix exponential, b = exp(a)

Referenced by c_to_d_zoh().

WrapPhase() [1/2]

double WrapPhase

(

double

angle

)

Wrap a phase angle to keep its value between -pi and +pi.

*sctlib

Parameters

angle

Phase angle [rad]

Returns

Phase angle wrapped between -PI and PI [rad]

References PI, and TWO_PI.

Referenced by add_elements(), CirclePhase(), El2Alfriend(), EllipsePhase(), FFEccGoals(), FFEccHills2Goals(), FFEccXExt(), FFEccYExt(), FFEccZExt(), GenerateTeamGoals(), InPlane(), MeanAnom2TrueLat(), MonitoringAlg(), OrbElemDiff(), orbit_acquisition_dv(), sub_elements(), and TargetTrueAnom().

WrapPhase() [2/2]

ml_matrix WrapPhase

(

ml_matrix

angle

)

Wrap a vector of phase angles to keep their values between -pi and +pi.

*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.

UnwrapPhase()

ml_matrix UnwrapPhase

(

ml_matrix

angle

)

Unwrap a vector of angular values so they change continuously instead of wrapping.

*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 diff(), PI, and TWO_PI.

Referenced by DeltaEl2AlfriendMat(), FFEccDiscreteHills(), GenerateTimeVector(), ImpulsiveLPManeuver(), LPEccentric(), LPEccentricGVE(), and MonitoringAlg().

diff()

ml_matrix diff

(

ml_matrix

a

)

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

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

Referenced by AccelVector2ManeuverStruct(), DeltaEl2AlfriendMat(), FFEccDiscreteHills(), FFEccEstimateCost(), FFEccXExt(), FFEccYExt(), FFEccZExt(), ImpulsiveLPManeuver(), LPEccentricGVE(), LPEccentricTimeWeight(), MonitoringAlg(), NearestOffset(), OptimalAssignment(), and UnwrapPhase().

delt_circ()

ml_matrix delt_circ	(	ml_matrix	x,
		ml_matrix	y
	)

Compute a delta in circular coordinates.

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, and pss_rem().

circle_fit()

ml_matrix circle_fit	(	ml_matrix	x,
		ml_matrix	y
	)

Compute a delta in circular coordinates.

Compute a delta in circular coordinates.

Parameters

x	column matrix
y	column matrix

Returns

s column matrix [a;b;r]

pinv()

ml_matrix pinv

(

ml_matrix

a

)

Pseudo inverse.

Pseudo inverse.

Parameters

a

matrix

Returns

orthogonal matrix

Referenced by az_el_range_range_rate_to_rv(), camera_calibration(), momentum_unloading(), and q_from_u().

downhill_simplex()

ml_matrix downhill_simplex	(	double	RHSconst ml_matrix &x, void *context,
		ml_matrix &	x,
		ml_matrix &	options,
		void *	context
	)

Downhill simplex.

Parameters

RHS	Function to minimize
x	Initial guess
options	Options
context	Context pointer to pass to RHS

Referenced by fit_PSF_to_data().

orthogonalize()

ml_matrix orthogonalize

(

ml_matrix

a

)

Orthogonalize.

Orthogonalize.

Orthogonalize a 3x3 matrix

Parameters

a

matrix

Returns

orthogonal matrix

Referenced by camera_calibration(), and q_from_u().

logistic() [1/2]

ml_matrix logistic

(

ml_matrix

x

)

Logistic function.

Parameters

x

input

Returns

1/(1+exp(-x))

logistic() [2/2]

double logistic

(

double

x

)

Logistic function.

Parameters

x

input

Returns

1/(1+exp(-x))

interp_extrap_1d()

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

1D interpolation and extrapolation

1D interpolation and extrapolation

Parameters

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

Returns

value

gaussian_multivariate_density()

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

Gaussian Mulivariate Normal Distribution function.

Gaussian Mulivariate Normal Distribution function.

Parameters

y	array of parameters
y_bar	corresponding means
P	covariance matrix

Returns

likelihood

roots_second_order() [1/2]

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

Roots second order.

Roots second order.

Parameters

a	Coefficient
b	Coefficient
c	Coefficient

Returns

roots

Referenced by roots_second_order().

roots_second_order() [2/2]

ml_matrix roots_second_order

(

const ml_matrix &

p

)

Roots second order.

Roots second order.

Parameters

p	array of coefficients

Returns

roots

References roots_second_order().

skew_symmetric()

ml_matrix skew_symmetric

(

const ml_matrix &

v

)

Generate a skew symmetric matrix.

Generate a skew symmetric matrix.

Parameters

v	Vector (3x1)

Returns

skew symmetric matrix

Referenced by attitude_propagation_discrete(), discrete_state_transition_matrix(), multi_body_central_hub_with_rwa::state_derivative(), and multi_body_central_hub::state_derivative().

trig_reduction()

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

Solve for beta: a*sin(beta)+b*cos(beta) = c.

Parameters

a	Coefficient
b	Coefficient
c	RHS

Returns

roots

m_k_3_by_3()

ml_matrix m_k_3_by_3	(	const ml_matrix &	m,
		int	k
	)

Returns a 3x3 matrix from a 3x3*n matrix.

Parameters

m	3x3*n matrix
k	index

Returns

3x3 matrix

Referenced by multi_body_central_hub_with_rwa::state_derivative(), and multi_body_central_hub::state_derivative().

intersect_line_sphere_true()

int intersect_line_sphere_true	(	const ml_matrix &	p1,
		const ml_matrix &	p2,
		double	r
	)

Find if a line intersects a sphere.

Find if a line intersects a sphere.

Parameters

p1	3x1 origin point on the sphere
p2	3x1 final point off the sphere
r	Radius of sphere

Returns

3x3 matrix

append_diagonal()

ml_matrix append_diagonal	(	const ml_matrix &	a,
		const ml_matrix &	b
	)

Append a diagonal matrix to another matrix.

Append a diagonal matrix to another matrix.

Parameters

a	square matrix
b	diagonal of a matrix

Returns

Combined matrix

trapezoidal_integration()

double trapezoidal_integration	(	double	Integranddouble x, void *context,
		double	a,
		double	b,
		int	n,
		void *	context
	)

Trapezoidal integration.

Trapezoidal integration.

Referenced by rhs_airy_defocused_psf().

two_dimensional_integration()

double two_dimensional_integration	(	ml_matrix	Integrandconst ml_matrix &x, const ml_matrix &y, void *context,
		double	a,
		double	b,
		double	c,
		double	d,
		int	nX,
		int	nY,
		void *	context
	)

Two dimensional integration.

Two dimensional integration.

Parameters

Integrand	pointer to the integrand function
a	Minimum x
b	Maximum x
c	Minimum y
d	Maximum y
nX	Number of divisions in x
nY	Number of divisions in y
*context	Pointer to data to pass to Integrand

Returns

Integral value

Referenced by imager_psf(), and imager_psf_uint16().