Transcript CHAPTER OBJECTIVES

CHAPTER
OBJECTIVES
The primary objective of this chapter is to provide you with a concrete idea of what
numerical methods are and how they relate to engineering and scientific problem solving.
Specific objectives and topics covered are
• Learning how mathematical models can be formulated on the basis of scientific
principles to simulate the behavior of a simple physical system.
• Understanding how numerical methods afford a means to generate solutions in a
manner that can be implemented on a digital computer.
• Understanding the different types of conservation laws that lie beneath the models used
in the various engineering disciplines and appreciating the difference between
steady-state and dynamic solutions of these models.
• Learning about the different types of numerical methods we will be covering in this
book.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to provide an introduction and overview of how
MATLAB’s command mode is used to implement interactive computations. Specific
objectives and topics covered are
• Learning how real and complex numbers are assigned to variables
• Learning how vectors and matrices are assigned values using simple
assignment, the colon operator, and the
and
functions.
• Understanding the priority rules for constructing mathematical expressions.
• Gaining a general understanding of built-in functions and how you can learn more about
them with MATLAB’s Help facilities.
• Learning how to use vectors to create a simple line plot based on an equation.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to learn how to write M-file programs to
implement numerical methods. Specific objectives and topics covered are
• Learning how to create well-documented M-files in the edit window and invoke them
from the command window.
• Understanding how to set up M-files so that they interactively prompt users for
information and display results in the command window.
• Learning how to write clear and well-documented M-files by employing structured
programming constructs to implement logic and repetition.
• Understanding what is meant by vectorization and why it is beneficial.
• Understanding how functions can be passed to M-files.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with the major sources of errors
involved in numerical methods. Specific objectives and topics covered are
• Understanding the distinction between accuracy and precision.
• Learning how to quantify error.
• Learning how error estimates can be used to decide when to terminate an iterative
calculation.
• Understanding how round-off errors occur because digital computers have a limited
ability to represent numbers.
• Recognizing that truncation errors occur when exact mathematical formulations are
represented by approximations.
• Knowing how to use the Taylor series to estimate truncation errors.
• Understanding how to write forward, backward, and centered finite difference
approximations of first and second derivatives.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with bracketing methods for
finding the root of a single nonlinear equation. Specific objectives and topics covered are
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Understanding what roots problems are and where they occur in engineering and science.
Knowing how to determine a root graphically.
Understanding the incremental search method and its shortcomings.
Knowing how to solve a roots problem with the bisection method.
Knowing how to estimate the error of bisection and why it differs from error estimates
for other types of root location algorithms.
• Understanding false position and how it differs from bisection.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with open methods for finding the
root of a single nonlinear equation. Specific objectives and topics covered are
• Recognizing the difference between bracketing and open methods for root location.
• Understanding the fixed-point iteration method and how you can evaluate its
convergence characteristics.
• Knowing how to solve a roots problem with the Newton-Raphson method and
appreciating the concept of quadratic convergence.
• Knowing how to implement both the secant and the modified secant methods.
• Knowing how to use MATLAB’s
function to estimate roots.
• Learning how to manipulate and determine the roots of polynomials with MATLAB.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with linear algebraic equations and
their relationship to matrices and matrix algebra. Specific objectives and topics covered are
• Understanding what linear systems of equations are and where they occur in engineering
and science.
• Understanding matrix notation.
• Being able to identify the following types of matrices: identity, diagonal, symmetric,
triangular, and tridiagonal.
• Knowing how to perform matrix multiplication and being able to assess when it is feasible.
• Knowing how to represent a system of linear algebraic equations in matrix form.
• Knowing how to solve linear algebraic equations with left division and matrix inversion in
MATLAB.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to describe the Gauss elimination algorithm for
solving linear algebraic equations. Specific objectives and topics covered are
• Knowing how to solve small sets of linear equations with the graphical method and
Cramer’s rule.
• Understanding how to implement forward elimination and back substitution as in Gauss
elimination.
• Understanding how to count flops to evaluate the efficiency of an algorithm.
• Understanding the concepts of singularity and ill-condition.
• Understanding how partial pivoting is implemented and how it differs from complete
pivoting.
• Recognizing how the banded structure of a tridiagonal system can be exploited to obtain
extremely efficient solutions.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with LU decomposition. Specific
objectives and topics covered are
• Understanding that LU decomposition involves factoring the coefficient matrix into two
triangular matrices that can then be used to efficiently evaluate different right-hand-side
vectors.
• Knowing how to express Gauss elimination as an LU decomposition.
• Given an LU decomposition, knowing how to evaluate multiple right-hand-side vectors.
• Recognizing that Cholesky’s method provides an efficient way to decompose a
symmetric matrix and that the resulting triangular matrix and its transpose can be used
to evaluate right-hand-side vectors efficiently.
• Understanding in general terms what happens when MATLAB’s backslash operator is
used to solve linear systems.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to show how to compute the matrix inverse and to
illustrate how it can be used to analyze complex linear systems that occur in engineering and
science. In addition, a method to assess a matrix solution’s sensitivity to round-off error is
described. Specific objectives and topics covered are
• Knowing how to determine the matrix inverse in an efficient manner based on LU
decomposition.
• Understanding how the matrix inverse can be used to assess stimulus-response
characteristics of engineering systems.
• Understanding the meaning of matrix and vector norms and how they are computed.
• Knowing how to use norms to compute the matrix condition number.
• Understanding how the magnitude of the condition number can be used to estimate the
precision of solutions of linear algebraic equations.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to acquaint you with iterative methods for solving
simultaneous equations. Specific objectives and topics covered are
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Understanding the difference between the Gauss-Seidel and Jacobi methods.
Knowing how to assess diagonal dominance and knowing what it means.
Recognizing how relaxation can be used to improve the convergence of iterative methods.
Understanding how to solve systems of nonlinear equations with successive substitution
and Newton-Raphson.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to introduce you to how least-squares regression can
be used to fit a straight line to measured data. Specific objectives and topics covered are
• Understanding the difference between regression and interpolation.
• Familiarizing yourself with some basic descriptive statistics and the normal distribution.
• Knowing how to compute the slope and intercept of a best-fit straight line with linear
regression.
• Knowing how to compute and understand the meaning of the coefficient of determination
and the standard error of the estimate.
• Understanding how to use transformations to linearize nonlinear equations so that they can
be fit with linear regression.
• Knowing how to implement linear regression with MATLAB.
CHAPTER OBJECTIVES
This chapter takes the concept of fitting a straight line and extends it to (a) fitting a
polynomial and (b) fitting a variable that is a linear function of two or more independent
variables. We will then show how such applications can be generalized and applied to a
broader group of problems. Finally, we will show how optimization techniques can be
used
to implement nonlinear regression. Specific objectives and topics covered are
• Knowing how to implement polynomial regression.
• Knowing how to implement multiple linear regression.
• Understanding the formulation of the general linear least-squares model.
• Understanding how the general linear least-squares model can be solved on MATLAB
with
either the normal equations or with left division.
• Understanding how to implement nonlinear regression with optimization techniques.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to introduce you to polynomial interpolation.
Specific objectives and topics covered are
• Recognizing that evaluating polynomial coefficients with simultaneous equations is an
ill-conditioned problem.
• Knowing how to evaluate polynomial coefficients and interpolate with MATLAB’s
and
functions.
• Knowing how to perform an interpolation with Newton’s polynomial.
• Knowing how to perform an interpolation with a Lagrange polynomial.
• Knowing how to solve an inverse interpolation problem by recasting it as a roots
problem.
• Appreciating the dangers of extrapolation.
• Recognizing that higher-order polynomials can manifest large oscillations.
CHAPTER OBJECTIVES
The primary objective of this chapter is to introduce you to splines. Specific objectives and
topics covered are
• Understanding that splines minimize oscillations by fitting lower-order polynomials to
data
in a piecewise fashion.
• Knowing how to develop code to perform a table lookup.
• Recognizing why cubic polynomials are preferable to quadratic and higher-order
splines.
• Understanding the conditions that underlie a cubic spline fit.
• Understanding the differences between natural, clamped, and not-a-knot end conditions.
• Knowing how to fit a spline to data with MATLAB’s built-in functions.
CHAPTER OBJECTIVES
The primary objective of this chapter is to introduce you to numerical integration. Specific
objectives and topics covered are
• Recognizing that Newton-Cotes integration formulas are based on the strategy of
replacing
a complicated function or tabulated data with a polynomial that is easy to integrate.
• Knowing how to implement the following single application Newton-Cotes formulas:
Trapezoidal rule
Simpson’s 1/3 rule
Simpson’s 3/8 rule
• Knowing how to implement the following composite Newton-Cotes formulas:
Trapezoidal rule
Simpson’s 1/3 rule
• Recognizing that even-segment–odd-point formulas like Simpson’s 1/3 rule achieve
higher
than expected accuracy.
• Knowing how to use the trapezoidal rule to integrate unequally spaced data.
• Understanding the difference between open and closed integration formulas.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to introduce you to numerical methods for
integrating given functions. Specific objectives and topics covered are
• Understanding how Richardson extrapolation provides a means to create a more
accurate
integral estimate by combining two less accurate estimates.
• Understanding how Gauss quadrature provides superior integral estimates by picking
optimal abscissas at which to evaluate the function.
• Knowing how to use MATLAB’s built-in functions
and
to integrate
functions.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to introduce you to solving initial-value problems
for
ODEs (ordinary differential equations). Specific objectives and topics covered are
• Understanding the meaning of local and global truncation errors and their relationship to
step size for one-step methods for solving ODEs.
• Knowing how to implement the following Runge-Kutta (RK) methods for a single ODE:
Euler
Heun
Midpoint
Fourth-order RK
• Knowing how to iterate the corrector of Heun’s method.
• Knowing how to implement the following Runge-Kutta methods for systems of ODEs:
Euler
Fourth-order RK
CHAPTER OBJECTIVES
The primary objective of this chapter is to introduce you to more advanced methods for
solving initial-value problems for ordinary differential equations. Specific objectives and
topics covered are
• Understanding how the Runge-Kutta Fehlberg methods use RK methods of different
orders
to provide error estimates that are used to adjust the step size.
• Familiarizing yourself with the built-in MATLAB functions for solving ODEs.
• Learning how to adjust the options for MATLAB’s ODE solvers.
• Learning how to pass parameters via MATLAB’s ODE solvers.
• Understanding the difference between one-step and multistep methods for solving
ODEs.
• Understanding what is meant by stiffness and its implications for solving ODEs.
CHAPTER
OBJECTIVES
The primary objective of this chapter is to introduce you to methods for solving eigenvalue
problems. Specific objectives and topics covered are
• Understanding how eigenvalues provide the solution for a special form of homogeneous
linear algebraic equations.
• Understanding how eigenvalue problems arise in engineering and scientific problems
dealing with vibrating and oscillating systems.
• Knowing how to determine and interpret eigenvectors.
• Knowing how to implement the polynomial method with MATLAB.
• Knowing how to implement the power method to determine either the highest or lowest
eigenvalue along with its accompanying eigenvector.
• Knowing how to determine eigenvalues and eigenvectors with the MATLAB
function.