Approximating values
An easy way to solve equations... or at least approximate the solution
3. Errors
5. Let's practise
1. Approximating
4 The bisection method
2. Iterations
Iterating
We could approximate the value of the square root of 45 as 6.5 and leave it that way but...
....if you calculate 6.52 the actual value is 42.25, which is not as close to 45 as we'd like, right? So, what could we do now...?
Repetition is the key to mastering... or in this case, to finding a more suitable value!
Solving equations
Can you find an approximation of the solution of this equation? What do we need to check first? How many iterations are necessary so that the error is smaller than 0.5?
Bisection method steps
The bisection method: approximating the solution of an equation
To solve an equation such as f(x) = 0
- Preliminary step: check if f(x) is continuous at [a,b]
- Middle point (tentative solution) xs=(a+b)/2
- Check whether f(a)f(xs)<0 or f(xs)f(b)<0
- If f(a)f(xs)<0 repeat steps 2-4 on interval [a,xs]If f(xs)f(b)<0 repeat steps 2-4 on interval [xs,b]
You may repeat these steps until the solution obtained is satisfactory, i.e., the error is smaller than the tolerance. Each repetition is called an iteration. And each iteration, divides the error by two. Initial error is and after n iterations, the error becomes
An iterative way of finding roots
Solving equations is not an easy task. Sometimes, the exact solution is not achievable or it takes too long to achieve to be worthy. In such cases, we can approximate the solution by using the bisection method! The bisection method is an iterative method aimed to solve equations in the form f(x) = 0 in wich the involved function, f(x), fulfils Bolzano's theorem. To apply the method, if f(x) is continuous, we simply need to find an interval [a,b] in wich the function changes sign and then proceed with the bisection method steps, until the approximation is good enough!
Errors are... unavoidable
The amount of numerical error that is acceptable varies depending on the problem we are facing. This acceptable amount is called tolerance of the error.
When approximating a value the way we are doing right now, we are bound to have a certain amount of numerical error. This does not mean that we are doing something wrong! Numerical errors are assumed whenever an approximation is made.
It is important to find the right balance between an acceptable numerical error and the advantages of approximating instead of calculating the exact value.
How would you...
calculate a square root without using a calculator?
Calculating a square root is not an easy feat. There is a method, but we tend to forget its steps, especially as we commonly use calculators instead. Think about this square root
Indeed! One way to approximate its value is remembering that the square root of 36 is 6 and the square root of 49 is 7. So the square root of 42 must be in between. Do you know why?
can you approximate its value without using any sort of electronic device?
Approximating values
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Transcript
Approximating values
An easy way to solve equations... or at least approximate the solution
3. Errors
5. Let's practise
1. Approximating
4 The bisection method
2. Iterations
Iterating
We could approximate the value of the square root of 45 as 6.5 and leave it that way but...
....if you calculate 6.52 the actual value is 42.25, which is not as close to 45 as we'd like, right? So, what could we do now...?
Repetition is the key to mastering... or in this case, to finding a more suitable value!
Solving equations
Can you find an approximation of the solution of this equation? What do we need to check first? How many iterations are necessary so that the error is smaller than 0.5?
Bisection method steps
The bisection method: approximating the solution of an equation
To solve an equation such as f(x) = 0
- Preliminary step: check if f(x) is continuous at [a,b]
- Middle point (tentative solution) xs=(a+b)/2
- Check whether f(a)f(xs)<0 or f(xs)f(b)<0
- If f(a)f(xs)<0 repeat steps 2-4 on interval [a,xs]If f(xs)f(b)<0 repeat steps 2-4 on interval [xs,b]
You may repeat these steps until the solution obtained is satisfactory, i.e., the error is smaller than the tolerance. Each repetition is called an iteration. And each iteration, divides the error by two. Initial error is and after n iterations, the error becomesAn iterative way of finding roots
Solving equations is not an easy task. Sometimes, the exact solution is not achievable or it takes too long to achieve to be worthy. In such cases, we can approximate the solution by using the bisection method! The bisection method is an iterative method aimed to solve equations in the form f(x) = 0 in wich the involved function, f(x), fulfils Bolzano's theorem. To apply the method, if f(x) is continuous, we simply need to find an interval [a,b] in wich the function changes sign and then proceed with the bisection method steps, until the approximation is good enough!
Errors are... unavoidable
The amount of numerical error that is acceptable varies depending on the problem we are facing. This acceptable amount is called tolerance of the error.
When approximating a value the way we are doing right now, we are bound to have a certain amount of numerical error. This does not mean that we are doing something wrong! Numerical errors are assumed whenever an approximation is made.
It is important to find the right balance between an acceptable numerical error and the advantages of approximating instead of calculating the exact value.
How would you...
calculate a square root without using a calculator?
Calculating a square root is not an easy feat. There is a method, but we tend to forget its steps, especially as we commonly use calculators instead. Think about this square root
Indeed! One way to approximate its value is remembering that the square root of 36 is 6 and the square root of 49 is 7. So the square root of 42 must be in between. Do you know why?
can you approximate its value without using any sort of electronic device?