ZOOMIN
Scalar Zero Finders

ZOOMIN is a FORTRAN90 library which seeks a root of a scalar function.

The library is based primarily on a book by Joseph Traub.

These routines are each intended to find one of more solutions of an equation in one unknown, written as

f(x) = 0

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

ZOOMIN is available in a FORTRAN90 version.

Related Data and Programs:

BISECTION_INTEGER, a FORTRAN90 library which seeks an integer solution to the equation F(X)=0, using bisection within a user-supplied change of sign interval [A,B].

BISECTION_RC, a FORTRAN90 library which seeks a solution to the equation F(X)=0 using bisection within a user-supplied change of sign interval [A,B]. The procedure is written using reverse communication.

BRENT, a FORTRAN90 library which contains Richard Brent's routines for finding the zero, local minimizer, or global minimizer of a scalar function of a scalar argument, without the use of derivative information.

TEST_ZERO, a FORTRAN90 library which defines functions which can be used to test zero finders.

ZERO_RC, a FORTRAN90 library which seeks solutions of a scalar nonlinear equation f(x) = 0, or a system of nonlinear equations, using reverse communication.

Reference:

1. Richard Brent,
   Algorithms for Minimization without Derivatives,
   Dover, 2002,
   ISBN: 0-486-41998-3,
   LC: QA402.5.B74.
2. Eldon Hansen, Merrell Patrick,
   A Family of Root Finding Methods,
   Numerische Mathematik,
   Volume 27, Number 3, September 1977, pages 257-269.
3. P Jarratt,
   Some fourth-order multipoint iterative methods for solving equations,
   Mathematics of Computation,
   Volume 20, Number 95, July 1966, pages 434-437.
4. Richard King,
   A family of fourth order methods,
   SIAM Journal on Numerical Analysis,
   Volume 10, 1973, pages 876-879.
5. Richard King,
   Improving the van de Vel root-finding method,
   Computing,
   Volume 30, 1983, pages 373-378.
6. Werner Rheinboldt,
   Algorithms for finding zeros of a function,
   UMAP Journal,
   Volume 2, Number 1, 1981, pages 43-72.
7. Joseph Traub,
   Iterative Methods for the Solution of Equations,
   Prentice Hall, 1964.
8. Hugo vandeVel,
   A method for computing a root of a single nonlinear equation, including its multiplicity,
   Computing,
   Volume 14, 1975, pages 167-171.

Source Code:

• zoomin.f90, the source code;

Examples and Tests:

• zoomin_test.f90, the calling program;
• zoomin_test.txt, the output file.

List of Routines:

• BISECT carries out the bisection method.
• BRENT implements the Brent bisection-based zero finder.
• CAP_PHI_03 implements the Traub capital Phi(0,3) method.
• CAP_PHI_21 implements the Traub capital PHI(2,1) function.
• CHEBYSHEV implements Chebyshev's method.
• DAGGER_E12 implements the dagger E 1,2 algorithm.
• E3 implements the Traub E3 method.
• E4 implements the Traub E4 method.
• EULER implements the Euler method.
• HALLEY1 implements Halley's method.
• HALLEY2 implements Halley's method, with finite differences.
• HALLEY_SUPER implements the super Halley method.
• HANSEN implements the Hansen and Patrick method.
• JARRATT implements the Jarratt fourth order method.
• JARRATT2 implements the inverse-free Jarratt fourth order method.
• KING implements a family of fourth order methods.
• LAGUERRE implements the Laguerre rootfinding method for polynomials.
• MIDPOINT implements the midpoint method.
• MULLER implements Muller's method.
• NEWTON implements Newton's method.
• NEWTON_MOD implements the modified Newton method.
• NEWTON_SEC implements the Newton - secant method.
• OSTROWSKI_SQRT implements the Ostrowski square root method.
• PERP_E_12 implements the Traub E 1,2 algorithm.
• PERP_E_21 implements the Traub perp E 21 method.
• PHI_12 implements the Traub capital PHI(1,2) method.
• PSI_21 implements the Traub PSI 2,1 method.
• PSI_12 implements the Traub PSI 1,2 method.
• R8_SWAP switches two real values.
• RED_CAP_PHI_04 implements the Traub reduced capital PHI(0,4) method.
• REGULA implements the Regula Falsi method.
• RHEIN1 implements the Rheinboldt bisection - secant method.
• RHEIN2 implements the Rheinboldt bisection - secant - inverse quadratic method.
• SCRIPT_E2 implements the Traub script E - 2 function.
• SCRIPT_E3 implements the Traub script E - 3 function.
• SCRIPT_E4 implements the Traub script E - 4 function.
• SECANT implements the secant method.
• SECANTX carries out the extended secant algorithm.
• STAR_E12 implements the Traub *E12 method.
• STAR_E21 implements the Traub *E21 method.
• STEFFENSON implements Steffenson's method.
• STIRLING implements Stirling's method.
• T14 implements the Traub fourteenth function.
• T15 implements the Traub fifteenth function.
• T16 implements the Traub sixteenth function.
• T_FAMILY1 implements the Traub first family of iterations.
• T_FAMILY2 implements the Traub second family of iterations.
• TE11F implements the Traub *E - 1,1(f) function.
• TE2U implements the Traub E - 2(u) function.
• TPHI1U implements the Traub phi - 1,1(u) function.
• TRAUB1 implements the Traub first method.
• TRAUB4 implements the Traub fourth method.
• TRAUB8 implements the Traub eighth function.
• TRAUB9 implements the Traub ninth method.
• TRAUB_OSTROWSKI implements the Traub-Ostrowksi method.
• TT1F implements the Traub type 1 functions 10 and 11.
• TTHIP implements the Traub third function.
• VANDEV implements the Van de Vel iteration.
• VANDEV2 implements the improved Van de Vel iteration.
• WHITTAKER implements the convex acceleeration of Whittaker's method.
• WHITTAKER2 implements the double convex acceleeration of Whittaker's method.
• ZOOMIN calls all the zero finders for a given problem.

You can go up one level to the FORTRAN90 source codes.