Minimisation and root-finding

CCATSL contains three functions for finding the minima and zeros of a real-valued function: MinCL finds the minima of a unimodal function in a specified interval, CubicRootsCL solves a general cubic equation and ZeroCL solves $f(x)$ = 0 in a given interval when $f$ is monotone.

Minimisation of a unimodal function, MinCL

MinCL finds the location of the minimum of a unimodal function in a given interval, using a method combining golden section search and parabolic interpolation.

	double MinCL (	double (*f)(double x),
		double a,
		double b,
		double tol);

	f		A user-defined function taking a single double argument and returning a double, giving the value of the function $f$ at the requested point.
	a		The lower endpoint of the search interval.
	b		The upper endpoint of the search interval.
	tol		The acceptable tolerance in the location of the minimum.

A typical use of MinCL might look like

/* /examples/chapter2/fminex.c */ #include <catam.h> double f(double x) { return 2.1+(x-2.3)*(x-2.3); } int MainCL(void) { double ans=MinCL(f,0,6,1e-5); printf("minimum occurs at %f",ans); return 0; }

Cubic equation solver, CubicRootsCL

CubicRootsCL solves the cubic equation $x^3$ + $ax^2$ + $bx$ + $c$ = 0. It returns the number of roots, and the roots in non-decreasing order.

	void CubicRootsCL (	double a,
		double b,
		double c,
		int *nroots,
		double *r1,
		double *r2,
		double *r3);

	a, b, c		The coefficients in the cubic equation $x^3$ + $ax^2$ + $bx$ + $c$ = 0.
	nroots		Pointer to a variable to hold the number of roots when CubicRootsCL returns.
	r1, r2, r3		Pointers to variables to hold the roots, in non-decreasing order.

Finding the zero of a function, ZeroCL

The CCATSL routine ZeroCL locates the root of an equation of the form $f(x)$ = 0 when $f$ is a monotone function, using a combination of bisection and more powerful methods.

	double ZeroCL (	double (*f)(double x),
		double a,
		double b,
		double tol);

	f		A user-defined function taking a double argument and returning a double, giving the value of the function $f$ at the requested point.
	a		The lower endpoint of the search interval.
	b		The upper endpoint of the search interval.
	tol		The acceptable tolerance in the location of the root.

Finding the roots of a polynomial, PolyRootsCL()

The CCATSL routine PolyRootsCL finds all the roots of a polynomial with real coefficients by employing the Laguerre's Method.

	void PolyRootsCL(	float *rcoeff,
		int deg,
		complex *roots,
		int polish);

	rcoeff		vector with the coefficients of the polynomial. These must be real.
	deg		degree of the polynomial.
	roots		complex array with the computed roots.
	polish		set to 1 for polishing the roots, 0 otherwise.