It 205 Numerical Analysis Question No. 1 What Is The Differe

It 205 Numerical Analysisquestion No 1 What Is The Difference Betwee

It 205: Numerical Analysis question no 1 . What is the difference between the Newton's method and the Secant method? Explain your answer using examples with MATLAB or/and Symbolab. question . What is the difference between an array, a matrix, and a vector? 2.

Answer the following questions for the array shown here. a) What is the size of c? (b) What is the value of c(2,3)? (c) List the subscripts of all elements containing the value 0.6. 3. Determine the size of the following arrays. Check your answers by entering the arrays into MATLAB and using the whos command or the Workspace Browser. Note that the later arrays may depend on the definitions of arrays defined earlier in this exercise. (a) u = [10 20*i 10+20]; (b) v = [-1; 20; 3]; (c) w = [1 0 -9; 2 -2 0; 1 2 3]; (d) x = [u' v]; (e) y(3,3) = -7; (f) z = [zeros(4,1) ones(4,1) zeros(1,4)']; (g) v(4) = x(2,1); 4.

What is the value of w(2,1) in the w array calculated in part (c)? 5. What is the value of x(2,1) in the x array calculated in part (d)? 6. What is the value of y(2,1) in the y array calculated in part (e)?

7. What is the value of v(3) after statement (g) is executed?

Paper For Above instruction

It 205 Numerical Analysisquestion No 1 What Is The Difference Betwee

It 205 Numerical Analysisquestion No 1 What Is The Difference Betwee

This paper explores foundational concepts in numerical analysis, particularly focusing on methodological differences and array manipulations within MATLAB. The first part examines the differences between Newton's Method and Secant Method, crucial root-finding techniques used for solving nonlinear equations. The second part delves into the distinctions between arrays, matrices, and vectors, followed by practical questions involving MATLAB array operations, indexing, and size determination.

Differences between Newton’s Method and Secant Method

Newton's method and the secant method are iterative algorithms used to find roots of nonlinear functions, but they differ significantly in their approach and requirements. Newton's method employs derivatives to approximate the root, using the tangent line to iteratively improve estimates. The formula is given by:

xn+1 = xn - f(xn) / f'(xn)

where f' denotes the derivative of the function. It converges rapidly when the initial guess is close to the actual root, especially if the function is smooth and differentiable around the root.

The secant method, on the other hand, does not require the explicit derivative but approximates it using two initial points, xn-1 and xn. Its iterative formula is:

xn+1 = xn - f(xn) * (xn - xn-1) / (f(xn) - f(xn-1))

This makes the secant method advantageous when derivatives are difficult to compute or not available, though it generally converges more slowly than Newton’s method.

Example in MATLAB:

% Newton's Method


x = 1; % initial guess


for i=1:10


x = x - f(x)/fprime(x);


end


% Secant Method


x0 = 0; x1 = 1;


for i=1:10


x2 = x1 - f(x1)*(x1 - x0)/(f(x1)-f(x0));


x0 = x1; x1 = x2;


end

Here, f(x) is the function whose root we seek, and fprime(x) its derivative. MATLAB's symbolic toolbox simplifies this process when derivatives are involved.

Difference between an Array, a Matrix, and a Vector

An array in MATLAB is a general term for a collection of elements, which can be multi-dimensional. Arrays can be one-dimensional (vector), two-dimensional (matrix), or higher dimensions. A vector is a one-dimensional array, either row or column. A matrix is a two-dimensional array with rows and columns.

Specifically:

  • A vector is a 1xN or Nx1 array.
  • A matrix is a 2D array with size MxN, where M and N are positive integers.
  • An array can also have three or more dimensions, extending beyond vectors and matrices for complex data structures.

Understanding these differences helps in effective data manipulation in MATLAB, especially during matrix operations and data analysis.

Array Manipulation and MATLAB Examples

Suppose we have an array c, and we need to analyze its size and contents. MATLAB functions like size, whos, and indexing facilitate this process.

For example, given an array c:

c = [0.5, 0.6, 0.4; 0.7, 0.6, 0.2];

To find the size of c:

size_c = size(c);

The value of c(2,3) is accessed directly, and all elements containing 0.6 can be identified via logical indexing:

indices = find(c == 0.6);


[row, col] = ind2sub(size(c), indices);

This demonstrates how array manipulations are performed in MATLAB for data analysis tasks.

Array Size and Content Evaluation

Using MATLAB, the sizes of various arrays can be checked with whos or size functions. The provided array definitions include complex scenarios such as creating new arrays through concatenation, transposition, and element assignment, highlighting MATLAB's flexibility.

For example:

  • Array u: u = [10 20*i 10+20]; has size 1x3.
  • Array v: v = [-1; 20; 3]; is a 3x1 column vector.
  • Array w: w = [1 0 -9; 2 -2 0; 1 2 3]; is a 3x3 matrix.
  • Array x: x = [u' v]; combines transposed u and v, resulting in a 3x2 array.
  • Array y: y(3,3) = -7; creates a 3x3 matrix with the element at (3,3) assigned to -7.
  • Array z: z = [zeros(4,1) ones(4,1) zeros(1,4)']; constructs a 4x3 array with mixed zeros and ones.
  • Array v: v(4) = x(2,1); assigns a new value to v's 4th element based on x.

Value Retrieval in Arrays

The specific array element values are obtained through indexing:

  • Value of w(2,1) in part (c):
  • From w, w(2,1) = 2.
  • Value of x(2,1) in part (d):
  • From x, x(2,1) = 20i, considering the transposition of u.
  • Value of y(2,1) in part (e):
  • Unspecified unless initialized, but if y exists, y(2,1) is the element in second row, first column.

Conclusion

Understanding the distinctions between root-finding algorithms like Newton's and Secant methods enhances numerical problem-solving skills. Additionally, proficient manipulation of arrays, matrices, and vectors in MATLAB is essential for data analysis, simulation, and scientific computing. Effective use of MATLAB functions for size determination, indexing, and array creation supports robust analytical workflows in numerical analysis.

References

  • Burden, R., & Faires, J. (2010). Numerical Analysis (9th ed.). Brooks/Cole.
  • Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers (7th ed.). McGraw-Hill Education.
  • Higham, N. J., & Higham, D. J. (2005). MATLAB Guide. SIAM.
  • Matlab Documentation. (2023). MATLAB Arrays and Matrices. MathWorks. https://www.mathworks.com/help/matlab/arrays-and-metrices.html
  • Meyer, C. D. (2000). Matrix Analysis and Applied Linear Algebra. SIAM.
  • Chapra, S. C. (2018). Numerical Methods for Engineers. Oxford University Press.
  • Kreyszig, E. (2011). Advanced Engineering Mathematics. Wiley.
  • Johnson, C. R. (2012). Numerical Methods and Analysis. Pearson.
  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
  • The MathWorks. (2023). MATLAB for Numerical Analysis. https://www.mathworks.com/products/matlab.html

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