C++ for Scientists - Technische Universität Dresden

14.5. THE BISECTION METHOD FOR FINDING THE ZERO OF A FUNCTION IN AN INTERVAL277
14.5.2 Software
The task is to first develop the function
template
void bisection( T& a, T& b, Function& f, double tau ) ;
that computes the bisection Algorithm 6. The object f is a functor that returns the function
value for a single argument x. The type T is a float type, i.e. float or double.
Write documentation for the function and describe the conceptual conditions on Function.
14.5.3 The growth and downfall of a caterpillar population
Everyone knows caterpillars grow up to be beautiful butterflies. But before they reach that
stage of their life, they need lots of food to grow. A large population will not grow at the same
rate as a smaller one, because of a shortage of food. Furthermore most birds enjoy a juicy
caterpillar as snack, so they are responsible for the premature death of several members of the
caterpillar population. These relationships can be modelled mathematically by the following
equation:
dN
dt
N αN 2
= rN(1 − ) −
K β + N 2
In this equation rN(1 − N
) models the growth of the population, where N equals the num-
K
ber of caterpillars, r is the growing rate of the population and K is the maximum amount of
caterpillars that can inhabit the area. The second term of the equation models the death of the
caterpillars. Here α is the maximum rate at which a bird can eat caterpillars when N is large
and β is a parameter that indicates the intensity of the bird attacks. We want to know when
there exists an equilibrium between the growth and death rate in the caterpillar population,
i.e. when dN
equals zero.
dt
Use the function bisection to compute the number of caterpillars N for which the following
populations are at an equilibrium in the intervals [0.1; 10], [10, 20] and [20, 100]:
• Population 1: r = 1.3, K = 100, α = 20 and β = 50
• Population 2: r = 2.0, K = 80, α = 25 and β = 10
Show the resulting roots in a table.
14.5.4 Computing eigenvalues using the bisection method
In this exercise, we use the function bisection to compute the eigenvalues of a real symmetric
dense matrix A with real eigenvalues. The problem is to compute λ so that
det(A − λI) = 0 .
The determinant is computed using the QR factorization (which is available in LAPACK). The
QR factorization computes an orthogonal matrix Q (Q T Q = I) and an upper triangular matrix
R so that
A − λI = QR .