ERG2012A: Advanced Engineering Mathematics (Syllabus B)L8 - Page 1 of 6Lecture 8: Numerical MethodsOutline•Basic concepts of numerical methods(Kreyszig: Section 17.1)§Real-number representation§Round-off§Loss of significant digits§Errors of numerical results§Error propagation§Algorithm•Solutions of equations by iteration(Kreyszig: Section 17.2)§Fixed-point iteration methodBasic Concepts of Numerical MethodsNumerical methods are used to solve problems on computers or calculators bynumerical calculations. The results are given as a table of numbers and/or graphicalrepresentations (figures).The steps from a given problem to the final answer are usually as follows:1.Setting up a mathematical model of the problem, such as a system ofequations, or a differential equation.2.Choosing mathematical method, perhaps together with preliminary errorestimation and a choice of step size.3.Programming, either using a computer languages, such as C and FORTRAN, orusing a computing environment, such as MATLAB, MAPLE, MATHEMATICA.4.Doing the computation.5.Interpreting the resultsin physical terms, including making decisions to rerun iffurther results are needed.Real-number RepresentationMany digital computers have two ways of representing numbers:(a)Fixed-point system, which represents all numbers with a fixed number of decimalplaces. For example, each of the numbers 62.358, 0.013, 1.000 has 3 decimalplaces.(b)Floating-point system, which keeps the number of significant digits fixed(whereas the decimal point is “floating”).Significant digitsof a numbercis any given digit ofc, except the zeros to the leftof the first nonzero digit that serve only to fix the position of the decimal point.For example, each of the numbers 1360, 1.360, 0.001360 has 4 significant digits.Also,026310623.8,106.238,100.0006238,100.6238××××have 4 significantdigits.Theoretically, we can represent any nonzero numberaas:is an integer.11.010emmae<≤⋅±=The fractional partmis calledmantissaandeis called theexponent.On the computer,mandeare limited. For example, the IEEE floating point standardfor single precision is about3838<<-eand for double precision308308<<-e.If in a computation a number outside that range occurs, this is calledunderflowwhenthe number is smaller andoverflowwhen it is larger.