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44 Chapter 3 – Numerical Solution Methods
3.2 Nonlinear equations and equation systems
Oneofthemostbasicnumericalproblemsencounteredincomputationaleconomicsisto
find the solution to a nonlinear equation or a whole system of nonlinear equations. A
nonlinear equation system usually can be defined by a function
f (x) : Rn → Rn
that maps an n dimensional vector x into the space Rn. We call the solution to the linear
equation system f(x)=0aroot of f. The root of a nonlinear equation system usually
has no closed-form solution. Consequently, various numerical methods addressing the
issue of root-finding were invented. In this section, we introduce a very simple one-
dimensional method called bisection search. As this method converges quite slow, we
show how to enhance speed by using Newton’s method. We then discuss some modi-
fications of Newton’s algorithm and introduce a solver for multidimensional nonlinear
equation systems. We close the chapter by showing the very general class of fixed-point
iteration methods. All those methods will, like in the previous section, be demonstrated
with a simple example.
Example Supposethedemandfunctioninagoodsmarketisgivenby
qd(p)=0.5p0.2+0.5p0.5,
where the first term denotes domestic and the second term export demand. Supply
should be inelastic and given by qs = 2 units. At what price p∗ does the market clear?
Setting supply equal to demand leads to the equation
0.5p0.2 +0.5p0.5 = 2
whichcanbereformulatedas
f (p)=0.5p0.2 +0.5p0.5 2 = 0. (3.2)
The second of the two equations exactly has the form f(p)=0 described above. The
market clearing price consequently is the solution to a nonlinear equation. Note that it is
not possible to derive an analytical solution to this problem. Hence, we need a numerical
methodtosolvefor p∗.
3.2.1 Bisection search in one dimension
Averyintuitive andad-hocapproachtosolvingnonlinearequationsinonedimensionis
theso-calledbisectionsearch. Thebasicideabehindthismethodisamathematicaltheorem
called intermediate value theorem. It states that, if [a,b] is an interval, f is a continuous
3.2. Nonlinear equations and equation systems 45
function and f(a) and f(b) have different signs, i.e. f(a) · f(b) < 0, then f has a root
within the interval [a,b]. The bisection algorithm now successively bisects the interval
[a, b] into intervals of equal size and tests in which interval the root of f is located by
using the above theorem.
Specifically, the algorithm proceeds as follows: suppose we had an interval [a ,b ] with
i i
the property that f(a )· f(b ) < 0, i.e. f has a root within this interval. We now bisect this
i i
interval into the sub-intervals [a ,x ] and [x ,b ] with
i i i i
a +b
xi = i i .
2
Wecan now test in which interval the root of f is located and calculate a new interval
[a , b ] by
i+1 i+1
a =a and b =x if f(a)· f(x ) < 0,
i+1 i i+1 i i i
a =x and b =b otherwise.
i+1 i i+1 i
Figure 3.1 demonstrates the approach.
f(x)
f(x)
b = b
i i+1
a x= a x* x
i ii+1
Figure3.1: Bisection search for finding the root of a function
Notethat,duetothebisectionof[a ,b ],wesuccessivelyhalvetheintervalsizewithevery
i i
iteration step. Consequently, xi converges towards the real root f(x∗) of f.Westop
the iteration process, if xi has sufficiently converged, i.e. the difference between two
successive values is smaller than an exogenously specified tolerance level ǫ
|x x|=|x a |=|x b |= 1 |b a |<ǫ.
i+1 i i+1 i+1 i+1 i+1 2i+2 0 0
46 Chapter 3 – Numerical Solution Methods
Program 3.5 shows how to find the equilibrium price in the above example by using
bisection search.
Program3.5: Bisection search in one dimension
program bisection
! variable declaration
implicit none
integer :: iter
real 8::x,a,b,fx,fa,fb
*
! set initial guesses and function values
a = 0.05d0
b = 0.25d0
fa = 0.5d0 a (-0.2d0)+0.5d0 a (-0.5d0)-2d0
* ** * **
fb = 0.5d0 b (-0.2d0)+0.5d0 b (-0.5d0)-2d0
* ** * **
! check whether there is a root in [a,b]
if(fa*fb >= 0d0)then
stop ’Error: There is no root in [a,b]’
endif
! start iteration process
do iter = 1, 200
! calculate new bisection point and function value
x = (a+b)/2d0
fx = 0.5d0 x (-0.2d0)+0.5d0 x (-0.5d0)-2d0
* ** * **
write(*,’(i4,f12.7)’)iter, abs(x-a)
! check for convergence
if(abs(x-a) < 1d-6)then
write(*,’(/a,f12.7,a,f12.9)’)’x=’,x,’ f=’,fx
stop
endif
! calculate new interval
if(fa fx<0d0)then
*
b=x
fb=fx
else
a=x
fa=fx
endif
enddo
write(*,’(a)’)’Error: no convergence’
end program
3.2. Nonlinear equations and equation systems 47
The program proceeds as follows: We first have to make an initial guess of the interval
in which the actual root of f is located. We chose a = 0.05 and b = 0.25 in our case.
0 0
Wethencalculate function values at the interval endpoints via (3.2) and check, whether
our condition for a root of f in [a ,b ] is satisfied. If not, we write an error message to
0 0
the console and abort the program. Having prepared all this, our iteration process starts
bycalculating x0. Note that we specify a maximum number of iterations in our do-loop.
If there is no convergence after 200 iteration, the program will be aborted with an error
message. Having calculated the new x and the respective function value, we can check
whether two successive iterates satisfy our convergence condition. If this is the case, we
stop the program and display our approximation of the root of f. If not, we choose the
appropriate interval and begin a new iteration.
Looking at the output of the program, we find that our tolerance level is halved in every
iteration step. This is obvious from
|x x|= 1 |b a |= 1 1 |b a |= 1|x x |.
i+1 i 2i+2 0 0 2 2i+1 0 0 2 i i1
In general, we call convergence speed linear, if
|xi+1 xi| < c|xi xi1| with c > 0,
i.e. the difference between the solutions of two successive iteration steps diminishes by
a constant factor c. Hence, bisection converges linearly to the actual root of f.Inthe
next section, we will show how to enhance this convergence speed by using some more
information about the shape of our function.
3.2.2 Newton’s method in one dimension
Oneofthebestknowandmostefficientmethodstosolvealinearequationistheso-called
Newton’smethod. Itisbasedontheideaofsuccessivelinearization,whichallowstoreplace
anonlinearroot-finding problemwithasequenceofsimplerlinearones. Thesolutionsof
the simpler problems finally converge to the solution of the nonlinear problem.
Starting at an arbitrary point x0, on which our function is defined, we can approximate f
byafirst-order Taylor expansion, i.e.
ˆ ′
f (x) ≈ f(x0)+f (x0)(x x0).
Asthefunctionvalue f(x0)andthederivativeof f at x0 usuallyareknown,wecaneasily
ˆ
findthesolution to the equation f(x)=0. This solution is given by
x =x f(x0).
1 0 f ′(x0)
Figure3.2demonstratestheapproach. Weapproximatethefunction f linearlyatthepoint
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