Difference between bisection and false position method
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The Bisection Method
The simplest way to solve an algebraic equation of the form g(z) = 0, for some function g is known as bisection. The method assumes that we start with two values of z that bracket a root: z1 (to the left) and z2 (to the right), say. Then, we iteratively narrow the range as follows. First, define zm to be the midpoint of the range: zm = (z1 + z2)/2. Then, if g(zm)has the same sign as g(z1), and assuming that there is only one root in the range, then we know that the root must lie in [zm,z2], and we can replace z1 by zm. If not, then the root lies in[z1,zm], so zm can replace z2.

A C fragment to implement this logic is:
if (g(zm)*g(z1) > 0) z1 = zm; else z2 = zm;
The process continues until |z2-z1|becomes less than some specified tolerance. Obviously, the uncertainty in the value of the root (the width of the range) decreases by a factor of two with each iteration, that is, we gain approximately 1 decimal digit of accuracy every 3 iterations.
Of course, finding the initial range [z1,z2] may not be a simple task. We will return to this later.
The False-Position and Secant Methods
The bisection method relies solely on the assumption that the function g is continuous, so its value at the midpoint (eventually) lies between its values at the end of the range. If g is differentiable, we can do better. In that case, a straight-line approximation to gin the range [z1,z2] is appropriate, and we can estimate the root by linear interpolation (or extrapolation). After each iteration, the range is refined to incorporate the improved information.

In C, we can write: zs = z1 + (z2 - z1) * (-g(z1)) / (g(z2) - g(z1));
Both the false-position and the secantmethods use this approach. The difference between the two is simply what you so with the information once you have it. In the method of false position (sometimes called regula falsi), we refine our range so that [z1,z2]always spans the root, as with bisection. That is, we replace z1 or z2 by zsdepending on the sign of g(zs):
/* False position iteration. */ zs = z1 + (z2 - z1) * (-g(z1)) / (g(z2) - g(z1)); if (g(zs)*g(z1) > 0) z1 = zs; else z2 = zs;
The simplest way to solve an algebraic equation of the form g(z) = 0, for some function g is known as bisection. The method assumes that we start with two values of z that bracket a root: z1 (to the left) and z2 (to the right), say. Then, we iteratively narrow the range as follows. First, define zm to be the midpoint of the range: zm = (z1 + z2)/2. Then, if g(zm)has the same sign as g(z1), and assuming that there is only one root in the range, then we know that the root must lie in [zm,z2], and we can replace z1 by zm. If not, then the root lies in[z1,zm], so zm can replace z2.

A C fragment to implement this logic is:
if (g(zm)*g(z1) > 0) z1 = zm; else z2 = zm;
The process continues until |z2-z1|becomes less than some specified tolerance. Obviously, the uncertainty in the value of the root (the width of the range) decreases by a factor of two with each iteration, that is, we gain approximately 1 decimal digit of accuracy every 3 iterations.
Of course, finding the initial range [z1,z2] may not be a simple task. We will return to this later.
The False-Position and Secant Methods
The bisection method relies solely on the assumption that the function g is continuous, so its value at the midpoint (eventually) lies between its values at the end of the range. If g is differentiable, we can do better. In that case, a straight-line approximation to gin the range [z1,z2] is appropriate, and we can estimate the root by linear interpolation (or extrapolation). After each iteration, the range is refined to incorporate the improved information.

In C, we can write: zs = z1 + (z2 - z1) * (-g(z1)) / (g(z2) - g(z1));
Both the false-position and the secantmethods use this approach. The difference between the two is simply what you so with the information once you have it. In the method of false position (sometimes called regula falsi), we refine our range so that [z1,z2]always spans the root, as with bisection. That is, we replace z1 or z2 by zsdepending on the sign of g(zs):
/* False position iteration. */ zs = z1 + (z2 - z1) * (-g(z1)) / (g(z2) - g(z1)); if (g(zs)*g(z1) > 0) z1 = zs; else z2 = zs;
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Step-by-step explanation:
Difference between bisection and false position method?
Difference between bisection and false position method:-
In bisection method an average of two independent variables is taken as next approximation to the solution while in false position method a line that passes through two points obtained by pair of dependent and independent variables is found and where it intersects abissica is takent as next approximation.
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