Question Analysis

Numerical Analysis:

Newton forward Interpolation(proof + math)*****

forward difference operator*

construct forward difference table**

Lagrange interpolation ****

Newton divided formula *

Euler (math + description+derivation)***

Modified Euler (how it improves accuracy)*

Taylor ** (skipped)

Range CUTE-AAAA**

absolute error+relative error**, overflow and underflow*

find percentage of error math*

chopping, general equation of chopping*

find absolute and relative error*

Numerical Differentiation * (skipped)

simson 1/3 ****

trapezoidal rule(def^n + math)***

Bisection (description, explanation, proof) ***

Newton raphson (math + equation)****

trancendental equation and characteristics (*)

Gauss method (describe)*

LU decomposition **

Curve(math + diff)**

numerical integration (proof+defn)* * (skipped )

find first and second derivative ***

iretative method*

gauss-seidel method*

gauss-jordan*

taxonomoy of error*

shifting operator basic

Errors:

Error, in applied mathematics, the difference between a true value and an estimate, or approximation, of that value .

Taxonomy of error:

*truncation

Absolute Error

Absolute error is the difference between measured and the actual value of a quantity.

Keyword: measured value - actual value

Relative Error

The relative error is defined as the ratio of the absolute error of the measurement to the actual measurement.

Keyword: absolute error/actual error

Rounding Error

Rounding error is the difference between a rounded-off numerical value and the actual value.

Percentage of Error

(Take absolute value)

Summary:

• Absolute Error = |Experimental Measurement – Actual Measurement|

• Relative Error= Absolute Error/Actual Measurement

• Percentage Error = Decimal Form of Relative Error x 100.

Truncation error

A truncation error is the difference between an actual and a truncated, or cut-off, value.

A truncated quantity is represented by a numeral with a fixed number of allowed digits, with any excess digits chopped off -- hence, the expression truncated

Example:

Consider the speed of light in a vacuum. The official value is 299,792,458 meters per second (m/s). In scientific (power-of-10) notation, it is expressed as 2.99792458 x 108 m/s. But truncating it to only two decimal places yields 2.99 x 108 m/s.

Since the truncation error is the difference between the actual value and the truncated value, in this case, it comes to the following:

2.99792458 x 108 - 2.99 x 108 = 0.00792458 x 108 m/s

Chopping Error:

• a type of round-off error

• truncated or chopping the last digit or last $k$ digit of a rounding value

(IT”S NOT A TURNCATION ERROR)

Transcendental Functions

The transcendental function can be defined as a function that is not algebraic and cannot be expressed in terms of a finite sequence of algebraic operations such as sin x.

Keyword: function which output is an infinite sequence

Transcendental equation

A transcendental equation is an equation into which transcendental functions (such as exponential, logarithmic, trigonometric, or inverse trigonometric) of one of the variables (s) have been solved for.

A transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function.[1]

Keyword: the equation which contains transcendental function

Characteristics (NOT SURE)

• non-algebraic

• infinite

• contains transcendental functions

(learn more : https://www.britannica.com/science/transcendental-function)

(If you are more interested about Errors, learn from : https://graphics.stanford.edu/courses/cs205a-13-fall/assets/notes/chapter1.pdf . Anyways, I haven’t read the PDF yet. All of them are collected from different articles.)

Bisection Method

The bisection method is an approximation method to find the roots of the given equation by repeatedly dividing the interval. This method will divide the interval until the resulting interval is found, which is extremely small.

More:

• used to find the roots of a polynomial equation

• based on intermediate theorem on continuous function

• work by narrowing the gap between positive and negative interval until closes in on the correct answer

Theory & Proof: (From Rupa)

Iterative Method

Newton Rapson Method

Theory & Proof : (from Rupa)

Shifting Operator

Forward Difference Table

Construct Forward Difference Table : (from Rupa)

Interpolation

In short, interpolation is a process of determining the unknown values that lie in between the known data points

x = 3 5 7 9 11

y= 2 6 8 12 13

The difference between $x{_i}$ and $x{_{i+1}}$ is equal. 2 for this example.

if,

x= 3.5 (at starting ) → we will use Newton Forward Interpolation

x= 5.1 - 7.9 → we will use Central Difference Interpolation (Out of syllabus)

x=9.5 (at end) → we will use Newton Backward Interpolation

Newton Forward Interpolation

Proof (from Rupa):

Math : (from Rupa)

Procedure:

Newton Backward Interpolation

less important

• starting from below

• ‘+’ sign instead of ‘-’

(start taking the value of y from bottom → lower value)

Missing Term in Interpolation

(YT: https://www.youtube.com/watch?v=P7fvPqdNOjM&ab_channel=B.K.TUTORIALS)

From Mishu:

Newton Divided Difference

What is Curve Fitting ?

(mark 4)

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points

Advantages:

• Simplicity: It is very easy to explain and to understand

• Applicability: There are hardly any applications where least squares doesn’t make sense

• Theoretical Underpinning: It is the maximum-likelihood solution and, if the Gauss-Markov conditions apply, the best linear unbiased estimator

Disadvantages/Drawbacks:***

• Sensitivity to outliers

• Test statistics might be unreliable when the data is not normally distributed (but with many datapoints that problem gets mitigated)

• Tendency to overfit data (LASSO or Ridge Regression might be advantageous)

• It can be quite sensitive to the choice of starting values.

• It is not readily applicable to censored data

Numerical Integration

Numerical Integration is a process of evaluating or obtaining a definite integral from a set of numerical values of the integrand f(x).

$h=\frac{b-a}{n}$

n = stripe, (generally, it’s 6. why 6 ? ⇒ because 6 is divisible by 2 and 3 ? Why 2 & 3 ? Check below ?)

Trapezoidal Rule

$\int_{a}^{b}{f(x)dx} = h(\frac{y_{0}+y{_n}}{2} + y{_1}....+ y{_n})$

Applicable for any no. interval.

Simpson one-third rule

$\int_{a}^{b}{f(x)dx} = \frac{h}{3}({y_{0}+y{_n}} + 4(y{_1}+y{_3}+...) + 2(y{_2}+y{_4}+...))$

Applicable for only even intervals.

Simpson three-by-eight rule

$\int_{a}^{b}{f(x)dx} = \frac{3h}{8}(y_{0}+y{_n} + 3(y{_1}+y{_2}+y{_4}+y{_5}...) + 2(y{_3}+y{_6}+...))$

Applicable for only multiple of 3 intervals.

WHY 6 ? ⇒ Because 6 is an even $\cap$ multiple of 3. And it’s the minimum which fill-up both conditions.

Lagrange Interpolation

The Lagrange interpolation formula is a way to find a polynomial which takes on certain values at arbitrary points

Approaches to Prove Lagrange:

LU Decomposition Factorization

To solve the Equation using LU Decomposition:

Least Square Method

(https://byjus.com/maths/least-square-method/#:~:text=The least square method is,the points from the curve.)

The least square method is the process of finding the best-fitting curve or line of best fit for a set of data points by reducing the sum of the squares of the points from the curve.

The equation of least square line is given by Y = a + bX

Normal equation for ‘a’:

∑Y = na + b∑X

Normal equation for ‘b’:

∑XY = a∑X + b∑X2

Question 1.: Find a straight line that fits the following data

Question 2 :

• y=something-something convert it to y=a+bx, and find new A, X, Y

• Create table

• Use those two formula and get the value of A,b … convert A to a.

From Fayaz:

Euler Method:

(https://www.youtube.com/watch?v=u5ggAyOOTUw&t=19s)

The Euler’s method is a first-order numerical procedure for solving ordinary differential equations (ODE) with a given initial value.

Euler proof (Approach) :

Modified Euler Method:

(https://www.youtube.com/watch?v=xLGDGeFZTnQ)

Instead of  approximating f(x, y) by  as in Euler’s method.  In  the  Modified Euler Method: we  have the iteration formula

Take $n=1$ to solve it by one step.

Accuracy

Euler method has a truncation error. To solve this problem modified Euler method is introduced. How ? →

• It takes arithmetic average of an interval (Xi, Xi+1) instead of a point. bla bla bla….

Range Kutta Method

(https://www.youtube.com/watch?v=fll1HdYy6vk&t=696s - 2nd Order)

(https://www.youtube.com/watch?v=JhI6cLRjKHY - 4th Order)

Runge–Kutta method is an effective and widely used method for solving the initial-value problems of differential equations. Runge–Kutta method can be used to construct high order accurate numerical method by functions' self without needing the high order derivatives of functions.

4th Order from Mishu:

Gauss Seidel Method

The Jacobi and Gauss-Seidel Iterative Methods

Iterative methods Jacobi and Gauss-Seidel in numerical analysis are based on the idea of successive approximations. This iterative method begins with one or two initial approximations of the roots, with a sequence of approximations x 1, x 2, x 3, ..., x k, ..., as k → ∞, this sequence of roots converges to exact root 𝛼.

https://byjus.com/maths/iterative-methods-gauss-seidel-and-jacobi/

Numerical Differentiation:

• Create Forward Difference Table

• Use the formulae for 1st, 2nd and 3rd derivative

Different between Gauss Elimination Method & Gauss Jordan Method

(https://www.geeksforgeeks.org/difference-between-gauss-elimination-method-and-gauss-jordan-method-numerical-method/)

In mathematics, the Gaussian elimination method is known as the row reduction algorithm for solving linear equations systems.

EXTRA:

Divided Difference Proof Idea and finding nth Divided Difference

Taylor:

(https://www.youtube.com/watch?v=82IDoaiYU0c&ab_channel=MKSTUTORIALSbyManojSir)