The GNSS integer ambiguities: estimation and validation

Choose the tolerance ε.
Determine the zero-crossing of the line connecting the limits of the interval:
xk+1 = xk −
xk − xk−1
f(xk) − f(xk−1) f(xk), k = 1, 2, 3, . . .
Compute the function value at xk+1, and evaluate whether or not:
|f(xk+1)| < ε
If yes, the root is found, otherwise continue.
A.3.3 False position method
The false position method works similar to the secant method, but instead of the most
recent estimates of the root, the false position method uses the most recent estimate
and the next recent estimate which has an opposite sign in the function value. The
procedure is then as follows.
Choose the interval [x − 0 , x+ 0 ] such that f(x−0 )f(x+ 0 ) < 0.
Choose the tolerance ε.
Determine the zero-crossing of the line connecting the limits of the interval:
xk+1 = x +
k −
x +
k
f(x +
k
− x−
k
) − f(x−
k )f(x+
k
), k = 0, 1, 2, . . .
Compute the function value at xk+1, and evaluate whether or not:
|f(xk+1)| < ε
If yes, the root is found, otherwise:
if f(xk+1)f(x +
k ) < 0 : x−
k+1 = xk+1, x +
k+1 = x+
k
else x −
k+1
and continue.
A.3.4 Newton-Raphson method
= x−
k
, x+
k+1 = xk+1
The Newton-Raphson method determines the slope of the function at the current estimate
of the root and uses the zero-crossing of the tangent line as the next reference
point. The procedure is as follows.
Choose the initial estimate x0.
Choose the tolerance ε.
Numerical root finding methods 149