Numerical Methods Charudatt Kadolkar 12/9/2018 Frequently, presenters must deliver material of a technical nature to an audience unfamiliar with the topic or vocabulary. The material may be complex or heavy with detail. To present technical material effectively, use the following guidelines from Dale Carnegie Training®.   Consider the amount of time available and prepare to organize your material. Narrow your topic. Divide your presentation into clear segments. Follow a logical progression. Maintain your focus throughout. Close the presentation with a summary, repetition of the key steps, or a logical conclusion. Keep your audience in mind at all times. For example, be sure data is clear and information is relevant. Keep the level of detail and vocabulary appropriate for the audience. Use visuals to support key points or steps. Keep alert to the needs of your listeners, and you will have a more receptive audience. Lectures on Numerical Methods 12/9/2018 Copyright 1996-98 © Dale Carnegie & Associates, Inc.

Interpolation Problem In general the function is known only for a few points or provided as a table. Example: A log table only mentions following values. x 1 1.5 2 3 3.5 4 Log(x) 0.0 0.17609 0.30103 0.47712 0.54407 0.60206 The problem is to find log(2.5). We may use a straight line interpolation between points 2 and 3. The straight line interpolation may be extended to a polynomial, since the straight line uses only two datapoints, when we have several data points available with us. We will find a polynomial which passes through all these points. In your opening, establish the relevancy of the topic to the audience. Give a brief preview of the presentation and establish value for the listeners. Take into account your audience’s interest and expertise in the topic when choosing your vocabulary, examples, and illustrations. Focus on the importance of the topic to your audience, and you will have more attentive listeners. Lectures on Numerical Methods 12/9/2018

Lectures on Numerical Methods Polynomials Forms of Polynomials If you have several points, steps, or key ideas use multiple slides. Determine if your audience is to understand a new idea, learn a process, or receive greater depth to a familiar concept. Back up each point with adequate explanation. As appropriate, supplement your presentation with technical support data in hard copy or on disc, e-mail, or the Internet. Develop each point adequately to communicate with your audience. Lectures on Numerical Methods 12/9/2018

Lectures on Numerical Methods Polynomials Nested Multiplication Algorithm Want to evaluate a polynomial P(x) given in Newton form, at x = z. bn = an For i = n – 1 to 0 step –1 bi = ai + ( z – ci+1 )* bi+1 Print the answer P(z) = b0. If you have several points, steps, or key ideas use multiple slides. Determine if your audience is to understand a new idea, learn a process, or receive greater depth to a familiar concept. Back up each point with adequate explanation. As appropriate, supplement your presentation with technical support data in hard copy or on disc, e-mail, or the Internet. Develop each point adequately to communicate with your audience. Lectures on Numerical Methods 12/9/2018

Lectures on Numerical Methods Polynomials Nested Multiplication Algorithm Example If you have several points, steps, or key ideas use multiple slides. Determine if your audience is to understand a new idea, learn a process, or receive greater depth to a familiar concept. Back up each point with adequate explanation. As appropriate, supplement your presentation with technical support data in hard copy or on disc, e-mail, or the Internet. Develop each point adequately to communicate with your audience. Lectures on Numerical Methods 12/9/2018

Lectures on Numerical Methods Polynomials To interpolate a function (unknown) whose values at n+1 distinct points are known, we will use a polynomial of degree <= n. Uniqueness If two polynomials P(x) and Q(x) of degrees less than or equal to k agree at k+1 distinct points, then P(x) and Q(x) are identical. Existence Lagranges polynomial given below is the proof of existence. If you have several points, steps, or key ideas use multiple slides. Determine if your audience is to understand a new idea, learn a process, or receive greater depth to a familiar concept. Back up each point with adequate explanation. As appropriate, supplement your presentation with technical support data in hard copy or on disc, e-mail, or the Internet. Develop each point adequately to communicate with your audience. Lectures on Numerical Methods 12/9/2018

Lagrange’s Polynomial Algorithm The data is given as ( xi , fi ). To calculate p(x) p = 0 For i = 1 to n t = 1 For j = 0 to n If j  i then t = t * ( x – xi ) / ( xi – xj ) p = p + t * fi P is the required value of p(x) If you have several points, steps, or key ideas use multiple slides. Determine if your audience is to understand a new idea, learn a process, or receive greater depth to a familiar concept. Back up each point with adequate explanation. As appropriate, supplement your presentation with technical support data in hard copy or on disc, e-mail, or the Internet. Develop each point adequately to communicate with your audience. Lectures on Numerical Methods 12/9/2018

Method of Divided Differences Determine the best close for your audience and your presentation. Close with a summary; offer options; recommend a strategy; suggest a plan; set a goal. Keep your focus throughout your presentation, and you will more likely achieve your purpose. Lectures on Numerical Methods 12/9/2018

Method of Divided Differences Table Determine the best close for your audience and your presentation. Close with a summary; offer options; recommend a strategy; suggest a plan; set a goal. Keep your focus throughout your presentation, and you will more likely achieve your purpose. Lectures on Numerical Methods 12/9/2018

Method of Divided Differences Algorithm To create the table of divided differences. For i = 1 to n di,0 = fi For j = 1 to n-i dj,i = (dj+1,i-1 – dj,i-1 ) / (xj+i – xj) D would be required table Determine the best close for your audience and your presentation. Close with a summary; offer options; recommend a strategy; suggest a plan; set a goal. Keep your focus throughout your presentation, and you will more likely achieve your purpose. Lectures on Numerical Methods 12/9/2018