Last Revision March, 2011
Approximation and Optimization
Seven topics are employed to introduce and explore concepts of numerical approximation and optimization.
The topics of this chapter can be accessed by clicking their names.
An Image from the Topic
Variable Step Size
Illustrates the efficacy of employing variable step size
Close-Bounds to the Ratio of the Circumference of a Circle to its Diameter and to the Ratio of a Circle's Area to the Square of its Radius
Two methods are shown and compared for bounding the value of π, the motivation being that up to this point only a lower bound had been established.
Design a Simple Eave Trough using Optimization
Introduces the use of Polar Coordinates in comparison with Rectangular Coordinates in the context of a simple optimization
Approximation with Orthogonal Functions using a Macro
Carries approximation concepts a step further by employing sinusoids in approximation. Introduces the use of a Spread Sheet Macro as an effective calculation method
Sensitivity in Approximation - Steepest Descent
Steepest Descent, a valuable method of optimization, is described and illustrated. An example macro is provided
Exploring a Limit with a Spreadsheet - A Use of e
A value for the constant e is derived. A use of e is then shown in the context of the comparison of the numerical solution of a simple differential equation with its analytic solution.
Irregular Repayments toward Loans - Irregular Returns from Investment - Continuous Compounding - A Use of e
Returns from Investments and repayments of loans often occur at irregularly spaced intervals and in non-uniform amounts.