04 - 4 Roundoff and Truncation Errors 4 Roundoff and...

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Unformatted text preview: 4: Roundoff and Truncation Errors 4: Roundoff and Truncation Errors Sep 29, 2006 10:02 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives To acquaint you with the major sources of errors involved in numerical methods. ● The distinction between accuracy and precision. ● The meaning of 'significant digits'. (n.i.t.) ● How to quantify error. ● How error estimates can be used to terminate iterative computations. ● Binary representation of numbers. (n.i.t.) ● Round-off errors because digital computers have limited ability to represent numbers. ● Truncation errors because exact mathematical formulations are represented by approximations. ● Use of the Taylor series to estimate truncation errors. ● Forward, backward and centred finite difference approximations of first and second derivatives. Sep 29, 2006 10:02 ECOR2606 -- Hassan & Holtz 3 Sources of Errors ● Roundoff errors : inability of computers (and calculators) to exactly represent certain numerical quantities. ● Truncation errors : representation of mathematical formulations by approximations (e.g., finite difference approximation, see below). ● Formulation or modelling errors: the mathematical model is a poor or incorrect model of the physical world (e.g, g is constant). ● Data uncertainty: mechanical or physical properties may not be known with sufficient accuracy (e.g. drag coefficient). ● Blunders: errors in calculation, recording, etc. Only the first two of these will be considered here. EG: the following finite-difference approximation to a first derivative was used to compute v(t i+1 ) in the bungee-jumper problem: dv dt ≈ v t = v t i 1 − v t i t i 1 − t i Sep 29, 2006 10:02 ECOR2606 -- Hassan & Holtz 4 4.1 Errors Accuracy – refers to how closely a computed or measured value agrees with the true value. Precision – refers to how closely repeated computed or measured values agree with each other. Figure 4.1: Inaccuracy ( bias ) – systematic deviation from the truth. Imprecision ( uncertainty ) – refers to the amount of “scatter”. Error – a term that refers to both. Sep 29, 2006 10:02 ECOR2606 -- Hassan & Holtz 5 Error Definitions True Error True Relative Error Approximate Relative Error True value = approximation error True relative error = true value − approximation true value a = approximate error approximation 100 % E t = true value − approximation t = true value − approximation true value 100 % a = present approximation − previous approximation present approximation 100 % 4.5 Sep 29, 2006 10:02 ECOR2606 -- Hassan & Holtz 6 Stopping Criteria Many numerical algorithms are iterations of the form: make initial estimate until error < maximum allowable error make an improved estimate ● do not know what the true answer is (obviously!)....
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This note was uploaded on 11/13/2011 for the course CIVE 2*** taught by Professor - during the Spring '11 term at Carleton CA.

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04 - 4 Roundoff and Truncation Errors 4 Roundoff and...

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