zoomin_report.txt  16 August 1998
 

ZOOMIN contains a large selection of scalar zero finders, for 
solving equations F(X)=0.  Most of these were collected from a 
book by Traub.
 

******************************************************************************80
 
To use ZOOMIN, write a main program which sets the input variables
and calls ZOOMIN as follows:
 
call ZOOMIN(a,b,c,abserr,kmax,idif,ipoly,mult,nsub)
 
the arguments have the following meanings-
 
a, b, c - three starting points, which should be close to
         the root for best results.  the program may rearrange
         these points so that f(a) is minimum and the sign of f(b)
         is opposite that of f(a) if possible.
 
abserr - an error tolerance.  if successive steps are of size
         less than abserr, or the function value of the current
         estimate is less than abserr, the program accepts
         the current estimate.
 
kmax   - the maximum number of iterations for any method.
         30 is a reasonable number.

idif   - the highest derivative you have supplied, with acceptable
         values from 0 to 3.  you must supply the function value
         (idif=0).  if you have supplied all derivatives from
         0 up to n, set idif=n.  whether or not you actually
         supply the correct derivatives, you must write the
         corresponding subroutines.  (see below).
 
ipoly  - the polynomial degree of the function, if it is a
         polynomial.  acceptable values are 2 or greater
         for a polynomial, or 1 or less for non polynomial.
         note that a linear function (ipoly=1) cannot
         be handled by laguerre's method.
  
mult   - the suspected multiplicity of the root, if known.
         acceptable values are 1 (for a single root), or
         greater.  for f(x)=(x-3)*(x-3)*(x-7), you should
         use mult=2 if you are seeking the root at x=3.
 
nsub   - the number of sub-iterations to take in the two
         methods which use them.  one method, for example,
         evaluates the derivative at a point, and then
         takes nsub newton-like steps, except that the
         derivative is held fixed for those nsub steps.
         acceptable values are 1 or greater.
 



*********************************************************************

Here is the output from a sample call to ZOOMIN:
  

 call zoomin to solve f(x)=(x+3)*(x+3)*(x-2)=0
  
  
 ZOOMIN
 A compilation of scalar zero finders,
 based on the work of Joseph Traub.
  
 1 point formulas use x1=   1.500000     fx1=  -10.12500    
 2 point formulas use x2=   4.000000     fx2=   98.00000    
 3 point formulas use x3=   1.000000     fx3=  -16.00000    
  
 User estimated multiplicity=           1
 Polynomial degree=           3
 Highest derivative supplied=           3
 Error tolerance=  9.9999997E-06
 Maximum number of steps=          30
 Newton method substep parameter =           3
  
  
 Technique                   Root               Steps
  
 1.  One point iteration functions with memory
  
Secant                            2.00000           6
Extended secant                   2.00000           6
Muller                            2.00000           5
Perp E 2,1                        2.00000           5
Star E 2,1                        2.00000           5
Finite difference Halley          2.00000           4
Phi 1,2                           2.00000           3
Perp E 1,2                        2.00000           3
Star E 1,2                        2.00000           3
Dagger E 1,2                      2.00000           3
  
 2. One point iteration functions.
  
Newton                            2.00000           4
E 3                               2.00000           3
E 4                               2.00000           3
Halley                            2.00000           3
Psi 2,1                           2.00000           2
Psi 1,2                           2.00000           3
Cap Phi 0,3                       2.00000           2
Reduced Cap Phi 0,4               2.00000           2
Ostrowski square root             2.00000           2
Euler                             2.00000           2
Laguerre                          2.00000           1
  
 3.  Multipoint iteration functions.
  
Traub first                       2.00000           4
Traub second                      2.00000           2
Traub twelfth                     2.00000           3
Traub thirteenth                  2.00000           2
Traub third, NSUB=  3             2.00000           3
Traub fourth, NSUB=  3            2.00000           2
Newton-secant                     2.00000           3
Traub sixth                       2.00000           3
Traub seventh                     2.00000           3
Traub eighth                      2.00000           3
Traub ninth                       2.00000           2
Traub type 1, form 10             2.00000           3
Traub type 1, form 11             2.00000           3
Traub fourteenth                  2.00000           2
Traub fifteenth                   2.00000           2
Traub sixteenth                   2.00000           2
King, BETA=0                      2.00000           2
King, BETA=1                      2.00000           2
King, BETA=2                      2.00000           3
Jarratt                           2.00000           2
  
 4.  Multiple root methods, multiplicity known.
  
Traub Script E 2                  2.00000           4
Traub Script E 3                  2.00000           3
Traub Script E 4                  2.00000           3
Traub Star E 1,1(f)               2.00000           6
  
 5.  Multiple root methods, multiplicity unknown.
  
Traub E 2(U)                      2.00000           4  mult=   1.000    
Traub Phi 1,1(U)                  2.00000           3  mult=   1.000    
Traub third                       2.00000           4  mult=   1.000    
Van de Vel                        2.00000           2  mult=   1.003    
Improved Van de Vel               2.00000           4  mult=   1.000    
  
 6. Bisection methods
  
Bisection                         2.00000          18
Regula falsi                      2.00000          20
Bisection-secant                  1.99998          11
Bisection-secant-inv quad         2.00000           5

 
*********************************************************************
 
A Compilation of Iterative Methods for the Solution of an Equation
in One Variable 

Harold A Deiss
 

Introduction
 
This compilation of iterative techniques to solve an equation in
one variable is a result of a National Science Foundation funded
program at the University of Pittsburgh.  This program enabled high
school teachers, such as myself, to share the work and experiences
involved in research.  It was my privilege to work under the guidance
of Professor Werner Rheinboldt whose inspiration, knowledge, and 
clarity of explanations made this compilation possible.  It is my 
hope that the algorithms and basic code may be useful to researchers
and practitioners in this field.
 

*********************************************************************
 
1) One point iteration functions with memory
 
*********************************************************************
 
1.1)  Secant method, PHI(1,1)
 

Comments: 

  Inverse interpolation.
  Traub also classifies this as Capital Phi(1,1), Perp E(1,1), Star E(1,1).  
  Its order is almost as high as Newton's method but needs no derivative.
 
Input:

  iterate(x)
  memory(x1,fx1)
  function(f)
 
Step: 

  fx := f(x)
  d1 := ( fx - fx1 ) / ( x - x1 )
  xn := x - fx / d1
 
Output:

  memory(x,fx)
  iterate(xn)
 
order=0.5*(1.0+sqrt(5)), roughly 1.62
 
 
1.2)  higher order secant method - Phi(2,1)
 
Traub, sections 4.22 and 6.12
 
Comments: 

  Inverse interpolation - 
  Phi(2,1) does not require the evaluation of derivatives; 
  Its use is indicated if the cost of evaluating f' is high.
 
Input:

  iterate(x)
  memory(x1,x2,fx1,fx2,d1)
  function(f)
 
Step:
  
  fx := f(x)
  d2 := d1
  d1 := (fx - fx1)/(x - x1)
  xn := x - fx/d1 + (fx*fx1/(fx-fx2))*(1.0/d1 - 1.0/d2)
 
Output:

  memory(x,x1,fx,fx1,d1)
  iterate(xn)
 
order=1.84
 
 
1.3) Phi(1,2) method
 
Traub, section 4.22 and 6.12
 
Comments: 
  Inverse interpolation;
  Although it uses no more information per step than the Newton method, 
  it enjoys a considerably higher order.
 
Input:
  
  iterate(x)
  memory(x1,fx1,dfx1)
  functions(f,f')
 
Step:   

  fx := f(x)
  dfx := f'(x)
  c := fx-fx1
  d := c/(x-x1)
  h := 1/c*(1/dfx - 1/d) - fx1/(c*c)*(1/dfx + 1/dfx1 -2/d)
  xn := x - fx/dfx + fx*fx*h
 
Output: 

  memory(x,fx,dfx)
  iterate(xn)
 
order=1 + sqrt(3), approximately 2.73
 
 
1.4)  Capital Phi(2,1)
 
Traub, sections 4.23 and 10.21
 
Comments: 

  Direct interpolation.
  This iterative function differs only in form from Muller's iterative 
  function over which it enjoys a number of advantages.  
  Its use is indicated if the cost of evaluating f' is high.
 
Input:  

  iterate(x)
  memory(x1,x2,fx1,fx2,d1)
  function(f)
 
Step:   

  fx := f(x)
  d2 := d1
  d1 := (fx-fx1)/(x-x1)
  d2 := (d1-d2)/(x-x2)
  z := d1 + (x-x1)*d2
  xn := x - 2.0*fx/(z + sqrt(z*z - 4.0*fx*d2))
 
Output: 

  memory(x,x1,fx,fx1,d1)
  iterate(xn)
 
order=1.84
 
 
1.5) perp E(2,1) method
 
Traub, sections 6.23 and 6.24
 
Comments: 

  Inverse interpolation in which f' is estimated.  
  Its use is indicated if the cost of evaluating f' is high.
 
Input:

  iterate(x)
  memory(x1,x2,fx1,fx2,d1)
  function(f)
 
Step:

  fx := f(x)
  d2 := d1
  d1 := (fx-fx1)/(x-x1)
  d := (fx-fx2)/(x-x2)
  xn := x-fx*(1/d1 + 1/d - 1/d2)
 
Output:

  memory(x,x1,fx,fx1,d1)
  iterate(xn)
 
order=1.84
 
 
1.6)  perp E(1,2) method
 
Traub, sections 6.23 and 6.24
 
Comments: 

  Inverse interpolation in which f" is estimated.
  Although it uses no more information per step than Newton's method,
  it enjoys a considereably higher order.
 
Input:

  iterate(x)
  memory(x1,fx1,dfx1)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d := (fx-fx1)/(x-x1)
  z := 2/dfx + 1/dfx1 - 3/d
  xn := x - fx/dfx + fx*fx*z/(fx-fx1)
 
Output:

  memory(x,fx,dfx)
  iterate(xn)
 
order=2.73
 
 
1.7)  star e 2,1 method
 
Traub, sections 6.22 and 6.24
 
Comments: 
  Inverse interpolation in which f' is estimated.  
  Its use is indicated if the cost of evaluation f' is high.
 
Input:  

  iterate(x)
  memory(x1,x2,fx1,fx2,d1)
  function(f)
 
Step:   

  fx := f(x)
  d2 := d1
  d1 := (fx-fx1)/(x-x1)
  d := (fx-fx2)/(x-x2)
  xn := x - fx/(d1 + d - d2)
 
Output: 

  memory(x,x1,fx,fx1,d1)
  iterate(xn)
 
order=1.84
 
 
1.8)  star e 1,2 method
 
Traub, sections 6.22 and 6.24
 
Comments: 

  Inverse interpolation in which f" is estimated.  
  Although it uses no more information per step than Newton's method, it 
  enjoys a considerably higher order.
 
Input:

  iterate(x)
  memory(x1,fx1,dfx1)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  d := (fx-fx1)/(x-x1)
  z := 2*dfx + dfx1 - 3*d
  xn := x - u - u*u*z/(dfx*(x-x1))
 
Output:

  memory(x,fx,dfx)
  iterate(xn)
 
order=2.73
 
 
1.9)   dagger e 1,2 method
 
Traub, section 6.33
 
Comments: 

  Inverse interpolation which estimates f".  
  This iterative function does not reuse the old information at fx1.  
  It is of somewhat simpler form than Star E 1,2 but has lower order.
 
Input:

  iterate(x)
  memory(x1,dfx1)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  d := (dfx-dfx1)/(x-x1)
  xn := x - u - 0.5*u*u*d/dfx
 
Output:

  memory(x,dfx)
  iterate(xn)
 
order=1+sqrt(2), approximately 2.41
 
 
1.10) finite difference halley's method
 
Traub, section 10.21
 
Comments: 

  Inverse interpolation.
  This is a finite difference analog of Halley's iterative function
 
Input: 

  iterate(x)
  memory(x1,x2,fx1,fx2,d1)
  function(f)
 
Step: 

  fx := f(x)
  d2 := d1
  d1 := (fx-fx1)/(x-x1)
  d2 := (d1-d2)/(x-x2)
  d := d1 - fx1*d2/d1
  xn := x - fx/d
 
Output:

  memory(x,x1,fx,fx1,d1)
  iterate(xn)
 
order=1.84
 
*********************************************************************
 
2)  one point iteration functions
 
*********************************************************************
 
2.1) Newton's method
 
Traub, sections 4.3 and 5.1
 
Comments: 

  Newton's method is also classified by Traub as Phi 0,2, Capital Phi 0,2, 
  and Psi 1,0.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  x := x - fx/dfx
 
Output:

  iterate(x)
 
order=2
 
 
2.2)  E3 method
 
Traub, sections 4.3 and 5.1
 
Comments: 

  Same as Phi 0,3, Psi 2,0.  
  E3 is particularly suitable if F satisfies a second order differential equation.
 
Input:

  iterate(x)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  x := x - u*(1.0 + v*u)
 
Output:

  iterate(x)
 
order=3
 
 
2.3)  e4 method
 
Traub, section 5.1
 
Comments: 

  same as Phi 0,4, Psi 3,0
 
Input:

  iterate(x)
  function(f,f',f",f"')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  d3fx := f"'(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  w := d3fx/(6*dfx)
  x := x - u*(1.0 + u*(v + u*(2*v*v - w)))
 
Output:

  iterate(x)
 
order=4
 
 
2.4)  Halley's method
 
Traub, sections 5.2 and 5.33; paper by g. alefeld
 
Comments: 

  Rational approximation.  Traub classifies as Psi 1,1.
 
Input: 

  iterate(x)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  x := x - u/(1.0 - v*u)
 
Output:

  iterate(x)
 
order=3
 
 
2.5)  Psi 2,1 method
 
Traub, section 5.2
 
Comments: 

  Rational approximation
 
Input:

  iterate(x)
  function(f,f',f",f"')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  d3fx := f"'(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  w := d3fx/(6*dfx)
  x := x - u*(v - (v*v - w)*u)/(v - (2*v*v - w)*u)
 
Output:

  iterate(x)
 
order=4
 
2.6)  Psi 1,2 method
 
Traub, sections 5.2 and 5.33
 
Comments: 

  Rational approximation
 
Input:

  iterate(x)
  function(f,f',f",f"')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  d3fx := f"'(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  w := d3fx/(6*dfx)
  x := x - u/(1.0 - u*(v + (v*v - w)*u))
 
Output:

   iterate(x)
 
order=4

 
2.7) Capital Phi 0,3
 
Traub, sections 5.31 and 5.32
 
Comments: 

  Direct interpolation
 
Input: 

  iterate(x)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  x := x - 2*u/(1.0 + sqrt(1.0 - 4*u*v))
 
Output:

  iterate(x)
 
order=3
 
 
2.8) reduced degree capital Phi 0,4
 
Traub, section 5.33
 
Comments: 

  Direct interpolation with Newton substitution
 
Input: 

  iterate(x)
  function(f,f',f",f"')
 
Step: 

  fx := f(x)
  dfx := f"(x)
  d2fx := f"(x)
  d3fx := f"'(x)
  u := fx/dfx
  v := d2fx/(2*dfx)
  w := d3fx/(6*dfx)
  x := x - 2*u/(1.0 + sqrt(1.0 - 4*u*(v - u*w)))
 
Output: iterate(x)
 
order=4
 
 
2.9)  a family of one point iteration functions
 
Eldon Hanson and Merrell Patrick,
A family of root finding methods,
numerical mathematics, volume 27, 1977.
 
Comments: 

  The general form is:

    xn=x-(beta+1)*f(x)/(beta*f'(x)+-sqrt((f'(x))**2-(beta+1)*f(x)*f"(x)))
    no method in class is asymptotically best for all f.
 
    typical values for the parameter beta are:
 
    Ostrowski:        beta=0
    Euler method:     beta=1
    Laguerre method:  if f is a polynomial of degree n, beta=1/(n-1)
    Halley method:    beta=-1
    Newton method:    beta tends to 0  (this method has a varying beta)
 
Input: 

  iterate(x)
  parameter(beta)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  x := x-(beta+1)*fx/(beta*dfx+sqrt(dfx*dfx-(beta+1)*fx*d2fx))
 
Output:

  iterate(x)
 
order=3 for simple roots for any fixed finite beta.
(order=2 for Newton's method).
 
*********************************************************************
 
3) multipoint iteration functions
 
*********************************************************************
 
3.1)  first function
 
Traub, sections 8.22, 8.4, and 9.31; see also page 236, function 1
 
Comments: 

  Uses chord at iteration point parallel to tangent line at Newton's point.
 
Input:

  iterate(x)
  function(f,f')
 
Step:

  fx := f(x)
  dfx := f'(x)
  z := x - fx/dfx
  dfz := f'(z)
  x := x - fx/dfz
 
Output:

  iterate(x)
 
order=3
 
 
3.2)  second function
 
Traub, section 8.34; see also page 236, function 3
 
Comments: 

  The case NSUB=1 is Newton's method.  
  The case NSUB=2 is derived in section 9.21.  
  The case NSUB=4 is derived in 9.22. 
  The iterative function is recursively formed.  This represents one 
  Newton step followed by NSUB-1 chord Newton steps with the same 
  derivative value.
 
Input:

  iterate(x)
  parameter(nsub)
  function(f,f')
 
Step: 

  z := x
  dfx := f'(x)
  DO I=1, nsub
    fz := f(z)
    z := z-fz/dfx
  end do
 
Output:

  iterate(z)
 
order=nsub+1
 
3.3)  a family of functions
 
Traub, sections 8.22 and 9.31; pages 236-237, functions 2, 12, 13
 
Comments: 

  Typical values for parameters c and d are:

    c = 1/2, d = 1, function 2 on page 236
    order=3
 
    c = 1/4, d = 2/3, function 12 on page 237
    order=3
  
    c = 5/12, d = 6/7, function 13 on page 237
    order=3
 
  Uses chord drawn at point on tangent line c*distance from
  original x and Newton iteration, parallel to tangent line at z.
 
Input:

  iterate(x)
  parameter(c,d)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  z := x - d*u
  dfz := f'(z)
  x := x - (c*u + (1 - c)*fx/dfz)
 
Output:

  iterate(x)
 
 
3.4)  fourth method
 
Traub, section 8.34; also see page 236, function 4.
 
Comments: 

  The iterative function is itself recursively formed.
 
Input:

  iterate(x)
  parameter(nsub)
  function(f,f',f")
 
Step:

  z := x
  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  do I=1,NSUB
    fz := f(z)
    z := z - fz/(dfx - d2fx*u)
  end do
 
Output:

  iterate(z)
 
order=2*nsub
c2=-f"/(2*f')
c3=c2*((-f*(f"'/f'-(f"/f')**2))/(2*f') + (c2)**2)
cj=((-f*(f"'/f'-(f"/f')**2))/(2*f'))(j-3)*c3, for j over 3
 
3.5)  Newton-secant iterative function.
 
Traub, sections 8.4 and 8.5; also see page 236, function 5
 
Comments: 

  Intersection with the x-axis of the secant line through
  (x,f(x)) and (x-u,f(x-u)).
 
Input: 

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  fxu := f(x - u)
  x := x - u + u*fxu/(fxu - fx)
 
Output:

  iterate(x)
 
order=3
 
 
3.6)  a family of multipoint iteration functions
 
Traub, section 8.4; see also page 236, functions 6 and 7.
 
Comments: 

  Typical values for the parameters a, b, c, and d are:
 
    a = 2, b = 3, c = 1, and d = 1, function 6 on page 236
    order=3
 
    a = 4, b = 7, c = 3, and d = 2/3, function 7 on page 236
    order=3
 
Input:

  iterate(x)
  parameter(a,b,c,d)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  dfxu := f'(x - d*u)
  x := x - (u/(a*dfx))*(b*dfx - c*dfxu)
 
Output: iterate(x)
 
3.7)  seventh function
 
Traub, section 8.4; see also page 237, function 8.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  dfxu := f'(x - (2/3)*u)
  x := x - 4*fx/(dfx + 3*dfxu)
 
Output:

  iterate(x)
 
order=3
 
 
3.8) eighth method
 
Traub, section 8.5; see also page 237, function 9.
 
Comments: 

  Iteration is of fourth order even though it uses but two
  evaluations of f and one of f'.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  fxu := f(x - u)
  x := x - u + u*fxu/(2*fxu - fx)
 
Output:

  iterate(x)
 
order=4
 
 
3.9)  type 1 functions
 
Traub, sections 9.21 and 9.22;
see also page 237, functions 10 and 11.
 
Comments: 

  Rho=0.5*(1-sqrt(5)).  Typical values for the parameter A are:
 
    a = 0, function 10 on page 237
    order=3
 
    a = 1, function 11 on page 237
    order=4
 
    constant: rho = 0.5*(1 - sqrt(5.0))
 
Input:

  iterate(x)
  parameter(a)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  fxu := f(x+rho*u)
  z := x-fxu/(rho*rho*dfx)
  fz := f(z)
  x := z-a*fz/dfx
 
Output:

  iterate(x)
 
 
3.10)  tenth function
 
Traub, section 9.32; see also page 237, function 14.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  dfxu := f'(x - u)
  z := x - 0.25*(u + fx/dfxu)
  dfz := f'(z)
  x := x - (1/6)*(u + fx/dfxu + 4*fx/dfz)
 
Output:

  iterate(x)
 
order=4
 
 
3.11)  eleventh function
 
Traub, section 9.32; see also page 238, function 15.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  dfxu := f'(x - u)
  z := x - (2/9)*(2*u + fx/dfxu)
  dfz := f'(z)
  x := x - 0.25*(u + 3*fx/dfz)
 
Output:

  iterate(x)
 
order=4
 
 
3.12)  twelfth function
 
Traub, section 9.32; see also page 238, function 16.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  dfxu := f'(x - (1/3)*u)
  z := x - 2*fx/(3*dfxu)
  dfz := f'(z)
  x := x - 0.25*(u + 3*fx/dfz)
 
Output:

  iterate(x)
 
order=4
 
 
3.13)  a family of fourth order methods
 
R F King,
a family of fourth order methods for nonlinear equations,
siam journal, vol 10, no. 5 (1973).
 
Comments: 

  Each method of the family is of fourth order and needs two
  function evaluations and one derivative per full step.
 
  Some common members of this family are with beta = 0, 1, and 2.
 
Input:

  iterate(x)
  function(f,f')
  parameter(beta)
 
Step: 

  fx := f(x)
  dfx := f'(x)
  w := x - fx/dfx
  fw := f(w)
  uw := fw/dfx
  x := w - uw*(fx + beta*fw)/(fx + (beta-2)*fw)
 
Output:

  iterate(x)
 
order=4
 
 
3.14)  fourteenth function
 
P Jarratt,
some fourth-order multipoint iterative methods for solving equations,
mathematics of computations, vol. 20, no. 95 (1966) pg. 434-437
 
Comments: 

  Computationally attractive methods in problems where the
  evaluation of f'(x) is rapid compared with f(x).  
  Such problems as when f(x) is defined by an integral.
 
Input:

  iterate(x)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  z := x - 2*u/3
  dfx := f'(z)
  x := x - u/2 + fx/(dfx - 3*dfz)
 
Output:

  iterate(x)
 
order=4
 
 
*********************************************************************
 
4) multiple roots, multiplicity given beforehand
 
*********************************************************************
 
4.1)  script e2 method
 
Traub, sections 7.3 and 7.4
 
Comments: 

  This is the well-known modification of Newton's method for
  multiplicity M greater than or equal to 1.
 
Input:

  iterate(x)
  parameter(m=multiplicity)
  function(f,f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  x := x - m*u
 
Output: 

  iterate(x)
 
order=2
 
 
4.2)  script e3 method
 
Traub, sections 7.3 and 7.4
 
Input:

  iterate(x)
  parameter(m=multiplicity)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  a2 := d2fx/(2*dfx)
  x := x - m*u*(0.5*(3 - m) + m*a2*u)
 
Output: 

  iterate(x)
 
order=3

 
4.3)  script e4 method
 
Traub, sections 7.3 and 7.4.
 
Input:

  iterate(x)
  parameter(m=multiplicity)
  function(f,f',f",f"')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  d3fx := f"'(x)
  u := fx/dfx
  a2 := d2fx/(2*dfx)
  a3 := d3fx/(6*dfx)
  x := x - m*u*((m*m - 6*m + 11)/6 + m*(2 - m)*a2*u +
                     m*m*(2*a2*a2 - a3)*u*u)
 
Output:

  iterate(x)
 
order=4
 
 
4.4)  *e1,1(f) method
 
Traub, section 7.6
 
Comments: 

  This algorithm behaves as though f were replaced by f**(1/m), 
  so a root of multiplicity m becomes a simple root.
  The method is then the secant algorithm on the new function.
 
Input:

  iterate(x)
  memory(x1,fx1)
  parameter(m=multiplicity)
  function(f)
 
Step: 

  fx := f(x)**(1/m)
  d := (fx-fx1)/(x-x1)
  xn := x - fx/d
 
Output:

  memory(x,fx)
  iterate(x)
 
order=0.5*(1+sqrt(5))
 
*********************************************************************
 
5)  multiple root, multiplicity to be determined
 
*********************************************************************
 
5.1)  e2(u) method
 
Traub, section 7.1; see also page 235, function 1.
 
Comments: 

  Newton's method with f replaced by U.  
  E2 requires evaluation of f, f', f" at each step.
 
Input:

  iterate(x)
  function(f,f',f")
 
Step: 

  fx := f(x)
  dfx := f'(x)
  d2fx := f"(x)
  u := fx/dfx
  m := (dfx*dfx)/(dfx*dfx-fx*d2fx)
  x := x-m*u
 
Output: 

  iterate(x)
 
order=2
 
 
5.2)  Phi 1,1 (u) method
 
Traub, section 7.6
 
Comments: 

  Same as Capital Phi 1,1(u).
 
Input:

  iterate(x)
  memory(u1)
  function(f, f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  fu := f(u)
  fu1 := f(u1)
  m := (u-u1)/(fu-fu1)
  x := x-m*u
 
Output:

  memory(u)
  iterate(x)
 
order=(1+sqrt(5))/2
 
 
5.3)  third function
 
Traub, section 7.8; see also page 235, function 3 at bottom.
 
Comments: 

  The order of this iterative function is incommensurate with our scale.  
  The quantity ln(f)/ln(u) converges to m.
 
Input:

  iterate(x)
  function(f, f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  m := ln//fx///ln//u//
  x := x - u*m
 
Output:

  iterate(x)
 
convergence analysis:  (ei+1/ei)*ln//ei// ~ - ln(m*//f(m)(6)/m!//1/m)
 
 
5.4) van de vel iteration
 
Comments: 

  If initial x is not close enough, m may become negative and diverge.  
  Since m = 1 is most frequent, initial m of one is usually chosen.
 
general root finding algorithm:
 
1)  perform Newton or secant to get initial close guess.
2)  iteration on x until m may be safely rounded to an integer.
3)  once m is found refine approximation of 6 to one of basic
iterations.
 
Input:

  iterate(x, m)
  function(f, f')
 
Step: 

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  z := x - m*u
  fz := f(z)
  dfz := f'(z)
  u1 := fz/dfz
  m := m*u/(u-u1)
  x := z - m*u1
 
Output:

  iterate(x,m)
 
order=1+sqrt(5)
 
 
5.5)  improved van de vel iteration
 
Comments: 

  One point method with memory.  
  Algorithm begins with one preliminary quasi-Newton step: x1 = x - m*u.
 
Input:

  iterate(x,m)
  memory(x1,u1)
  function(f, f')
 
Step:

  fx := f(x)
  dfx := f'(x)
  u := fx/dfx
  m := m*u1/(u1-u)
  xn := x - m*u
 
Output:

  memory(x,u)
  iterate(x,m)
 
order=(1+sqrt(5))/2
 
*********************************************************************
 
6) bisection methods
 
*********************************************************************
 
6.1) bisection
 
Comments: 

  Requires a change of sign interval.  
  Guaranteed convergence, but requires K steps to get K accurate binary digits.
 
Input:

  iterates(xpos,xneg)
  function(f)
 
Step:

  step=0.5*(xpos-xneg)
  xnew=xneg+step
  fx=f(xnew)
  if(fx.gt.0.0)then
    xpos=xnew
  else
    xneg=xnew
  endif
 
Output:

  iterates(xpos,xneg)
 
6.2) regula falsi
 
Comments: 

  Requires a change of sign interval.  
  Guaranteed convergence, but it may be very slow, slower than bisection.
 
Input:

  iterates(xpos,xneg)
  function(f)
 
Step: 

  step=-f(xneg)*(xpos-xneg)/(f(xpos)-f(xneg))
  xnew=xneg+step
  fx=f(xnew)
  if(fx.gt.0.0)then
    xpos=xnew
  else
    xneg=xnew
  endif
 
Output:

  iterates(xpos,xneg)
 
6.3) rheinboldt bisection-secant
 
Comments: 

  Requires a change of sign interval.  
  Combines the guaranteed convergence of bisection with the speed of the
  secant method.
 
Input:  

  iterates(a,b)
  function(f)
  tolerance(abserr)
 
init  : i=0
  t=0.5*abs(a-b)
  c=b
 
Step:   if(abs(f(a)).gt.abs(f(b))then
    c=a
    a=b
    b=c
  em=0.5*(b-a)
  if(abs(em).le.abserr)done
  p=(a-c)*f(a)
  q=f(c)-f(a)
  if(p.lt.0)then
    p=-p
    q=-q
  c=a
  i=i+1
  if(i.le.3)go to 20
  if(8*abs(em).gt.t)then
    a=a+em
    go to 22
  else
    i=0
    t=em
  if(p.le.abs(q)*tol)then
    a=a+sign(em)*abserr
    go to 22
     20 if(p.lt.q*em)then
    a=a+p/q
  else
    a=a+em
  if(sign(f(a)).eq.sign(f(b))then
    b=c
     22 done step
 
Output: 

  iterates(a,b)
 
6.4) rheinboldt bisection-secant-inverse quadratic
 
Comments: 

  Requires a change of sign interval.  
  Combines the guaranteed convergence of bisection with the speed of the
  secant and inverse quadratic methods.
 
Input:

  iterates(a,b)
  function(f)
  tolerance(abserr)
 
init  : i=0
  t=0.5*abs(a-b)
  c=b
 
Step:   if(abs(f(a)).gt.abs(f(b))then
    c=a
    a=b
    b=c
  em=0.5*(b-a)
  if(abs(em).le.abserr)done
 
  set up secant or regula falsi step
 
  if(2*abs(c-a).lt.abs(b-a))then
    ps=(a-c)*f(a)
    qs=f(c)-f(a)
  else
    ps=(a-b)*f(a)
    qs=f(b)-f(a)
  if(ps.lt.0)then
    ps=-ps
    qs=-qs
 
  set up inverse quadratic step
 
  piq=0.0
  qiq=0.0
  if(b.ne.c)then
    u=f(a)/f(c)
    v=f(c)/f(b)
    w=f(a)/f(b)
    piq=u*(2.0*em*v*(v-w)-(a-c)*(w-1.0))
    qiq=(u-1.0)*(v-1.0)*(w-1.0)
    if(piq.gt.0.0)qiq=-qiq
    piq=abs(piq)
 
  save old minimal point, check for forced bisection
 
  c=a
  i=i+1
  if(i.le.3)go to 20
  if(8*abs(em).gt.t)then
    a=a+em
    go to 22
  else
    i=0
    t=em
 
  choose step of highest order that stays within
  interval and is not too large or small
 
  stpmin=(abs(a)+abs(em)+1)*abserr
     20 if(piq.lt.1.5*em*qiq.and.abs(piq).gt.abs(qiq)*stpmin)then
    step=piq/qiq
  else if(ps.lt.qs*em.and.abs(ps).gt.abs(qs)*stpmin)then
    a=a+ps/qs
  else
    a=a+em
  if(sign(f(a)).eq.sign(f(b))then
    b=c
     22 done step
 
Output:

  iterates(a,b)