MATH3820 Assignment 1 Sem 1, 2017

Important Information

• Total mark= 30, assignment counts for 15% of final grade as per course outline.

• Due date: this assignment is due at 5pm on thursday April 13. You can submit either to the lecturer or to the maths office V123.

• This is an individual assignment. While collaboration is encouraged to stimulate learning and understanding, each student is responsible for understanding any solution submitted under his or her name and must write up the solutions independently.

• The submission must be accompanied with the signed cover sheet downloaded from http://www.newcastle.edu.au/__data/assets/pdf_file/0008/75383/AssessmentItemCoverSheet.

pdf.

• Include the print-out of your graphs in the submission.

• Always make and keep a copy of your work for your own records.

• Do not provide only the final answers. Show all your calculations and give explanations as appropriate. Write neatly. Include at least four digits after the decimal point in your calculations and results. You can use the matlab backslash operator to solve a linear system of equations in matlab.

Problems

1. Use 4-digit rounding to estimate the value of

and compute the relative error in the calculation.

[4 marks] 2. Consider the quadratic equation

which has roots and .

(a) Use 3-digit chopping to estimate the value of x1 and x2 and compute the relative errors of these estimates.

MATH3820 Assignment 1 Sem 1, 2017

(b) An alternative form for x2 is

. (1)

Use 3-digit chopping to estimate the value of x2 using equation (1) and compute the relative error of the estimate.

[5 marks] 3. Let f(x) = cosx.

(a) By considering an appropriate Vandermonde matrix, find the interpolation polynomial p3 of degree 3 that interpolates the values of f at the points x = 0,p/4,p/2,p.

(b) Use matlab to plot y = cosx and y = p3(x) on the interval [0,p] on the same graph and include the graph in your submission. (c) Use the error bound formula to find a bound for

.

(d) Use the matlab function max to estimate the value of E.

(e) Compute the Lagrange form of the interpolating polynomial p3.

(f) Compute the Newton form of the interpolating polynomial p3.

[7 marks]

4. Consider the following tabulated values of a function f(x) and its derivatives f0(x) at three points:

x f(x) f0(x)

0 1 2

1 1 -1

2 1 -1

(a) Compute the Hermite interpolating polynomial for this data.

(b) Use matlab to plot the interpolating polynomial on the interval [0,2] and include the graph in your submission. Use your answer to estimate the values of f(3/2) and f0(3/2).

[7 marks] 5. Construct a natural cubic spline interpolant for the data

x 0 1 2

y 1 1 2

Use matlab to plot the spline and include the graph in your submission. Find the first and second derivative of the cubic spline at x = 1. Is the spline three times differentiable at x = 1? [7 marks]

Important Information

• Total mark= 30, assignment counts for 15% of final grade as per course outline.

• Due date: this assignment is due at 5pm on thursday April 13. You can submit either to the lecturer or to the maths office V123.

• This is an individual assignment. While collaboration is encouraged to stimulate learning and understanding, each student is responsible for understanding any solution submitted under his or her name and must write up the solutions independently.

• The submission must be accompanied with the signed cover sheet downloaded from http://www.newcastle.edu.au/__data/assets/pdf_file/0008/75383/AssessmentItemCoverSheet.

pdf.

• Include the print-out of your graphs in the submission.

• Always make and keep a copy of your work for your own records.

• Do not provide only the final answers. Show all your calculations and give explanations as appropriate. Write neatly. Include at least four digits after the decimal point in your calculations and results. You can use the matlab backslash operator to solve a linear system of equations in matlab.

Problems

1. Use 4-digit rounding to estimate the value of

and compute the relative error in the calculation.

[4 marks] 2. Consider the quadratic equation

which has roots and .

(a) Use 3-digit chopping to estimate the value of x1 and x2 and compute the relative errors of these estimates.

MATH3820 Assignment 1 Sem 1, 2017

(b) An alternative form for x2 is

. (1)

Use 3-digit chopping to estimate the value of x2 using equation (1) and compute the relative error of the estimate.

[5 marks] 3. Let f(x) = cosx.

(a) By considering an appropriate Vandermonde matrix, find the interpolation polynomial p3 of degree 3 that interpolates the values of f at the points x = 0,p/4,p/2,p.

(b) Use matlab to plot y = cosx and y = p3(x) on the interval [0,p] on the same graph and include the graph in your submission. (c) Use the error bound formula to find a bound for

.

(d) Use the matlab function max to estimate the value of E.

(e) Compute the Lagrange form of the interpolating polynomial p3.

(f) Compute the Newton form of the interpolating polynomial p3.

[7 marks]

4. Consider the following tabulated values of a function f(x) and its derivatives f0(x) at three points:

x f(x) f0(x)

0 1 2

1 1 -1

2 1 -1

(a) Compute the Hermite interpolating polynomial for this data.

(b) Use matlab to plot the interpolating polynomial on the interval [0,2] and include the graph in your submission. Use your answer to estimate the values of f(3/2) and f0(3/2).

[7 marks] 5. Construct a natural cubic spline interpolant for the data

x 0 1 2

y 1 1 2

Use matlab to plot the spline and include the graph in your submission. Find the first and second derivative of the cubic spline at x = 1. Is the spline three times differentiable at x = 1? [7 marks]

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