more advanced techniques to solve mathematical problems to obtain an exact solution
Analytical Solution
Math problem is reformulated so it can be solved by arithmetic operations to obtain its approx. solutions
Numerical Solution
A source of error rooting from limited significant figures to represent exact num
Round-off error
A source of error from truncating or approximating a mathematical procedure
truncation Error
A source of error through mathematical functions
Propagation error
A source of error through mathematical functions
Propagation error
A source of error through mathematical functions
Propagation error
A source of error through mathematical functions
Propagation error
Converting from Binary to decimal
- Get sign bit by:
1- Negative
0- positive - Get mantissa by
Adding 0. to the leftmost mantissa
Adding 1 to mantissa - Converting mantissa from binary to decimal
Get exponent by
Converting exponent to decimal - Get e bias
2^(exponent bits-1) - 1 - Plug in to the formula and compute
s*m base 10 * 2^ (e base 10- e bias)
Converting from Decimal to binary
- Get sign bit
+ = 0- = 1
- Convert Decimal to binary
- Normalize to scientific notation to get the initial mantissa and the unbiased exponent
Get the exponent of the base two to get unbiased exponent - Get the mantissa by
subtracting 1. to the equation
Convert decimal to binary - Get the exponent of base 2 by:
Adding 127 to unbiased exponent
Convert decimal to binary
Two Categories of Roots of an Equation
Bracketing Methods
Open Methods
This method is where the root is located within an interval prescribed by a lower and an upper bound. Repeated application of these methods always results in closer estimate of the true value of the root
Bracketing Methods
Bracketing methods includes ___ and __ methods
bisection and false-position
This method of finding roots of an equation require only a single starting value of x or two starting values that do not necessarily bracket the root
Open Methods
Open methods includes __ and ___
Newton-Rapson and Secant methods
___ is a bracketing method for finding roots of an equation that can be done by getting the midpoint of the boundary continuously until the midpoint will approximately or equal to zero when plugin to the function f(x)
Bisection method
How to do Bisection method:
- Check whether point A and point b have opposite values when plugged in to f(x)
- If yes, then get the midpoint between the two points
- Plugin the midpoint to f(x). If the answer is closer to zero, use that as the new boundary.
4 Verify the boundary by using comparisons (<>=) - Repeat process from 1 to 4 until the value of f(x) is approx or equal to zero or to the specified value
How to to False-Position Method
- Get the f(x)’s of the roots a and b
- Subs the values to the formula to get bn or the next point
- Subs f(bn)
- If the f(bn) is positive, change a to bn
If f(bn) is negative, change b to bn
How to do newton-raphson Method
- Guess a point and sub f(guess)
If f(guess) is not = 0, continue - Derive f(x) then f’(guess)
- Subs to formula
- Repeat Process
How to do secant method
- Guess two points
2.Compute f(guess1) and f(guess 2) - Subs to formula
- Repeat until the new point is approx zero
Reason for roots to fail:
The point give a __ slope
Zero slope
Reason for roots to fail:
The consecutive points is in ___ of a function
Symmetry
