ASE311NOTES_Exam2 - ASE 311 Notes Chapter 4 Round-off and Truncation Errors Sources of Error 1 Human Error 2 Model Error you cannot consider everything

ASE311NOTES_Exam2 - ASE 311 Notes Chapter 4 Round-off and...

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ASE 311 Notes Chapter 4: Round-off and Truncation Errors Sources of Error 1. Human Error 2. Model Error: you cannot consider everything that can happen 3. Data Uncertainty: noise, repeatability (possible to set up experiment exactly the same everytime?), etc 4. Machine/Numerical Error: round-off error, Ex) pi is digit numbers Error Characterizations if there exists a true value, that true value equals approximation + error Accuracy: how closely a computed or measured value agrees with the true value Precision: how closely individual computed values agree with each other * we want everything to be both accurate and precise Example Case 1: true = 1, approx. = 1.1; .1 Case 2: true = 10,000, approx. = 10,000.1 .1 * we would call Case 2 more accurate because relative to true value, the error is considerably smaller *Never have true value Use for the stopping condition for iterative methods Numerical Error 1. Round-Off: exists because computers cannot represent numbers exactly a) computers have size and precision limits on their ability to represent numbers Computers represent numbers with a binary (base-2) system; this system has 0 or 1, or off/on respectively
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Example: 101.1 in base-2 = in base-10 Floating-Point Representation , MATLAB has 64 bits; 1 for the sign, 11 for the exponent, and 52 for the mantissa More bits if more bits are given to the mantissa, representation can be more accurate if more bits are given to the exponent, representation can include larger numbers Since numbers get normalized, in base-2, this implies that you get “53” bits in the mantissa Range : Largest Value: Smallest Value: Precision : *We cannot control round-off error as much as Truncation error 2. Truncation : errors that result from using an approximation in place of an exact mathematical procedure Example this is an approximation of a value that could be found using an exact mathematical procedure Taylor’s Theorem/Taylor’s Series Taylor’s Series Expansion As , this is exact, but since you do not go to , you have error (remainder term, ) The term is commonly represented in Lagrange form, where The Lagrange form comes from the mean value theorem of integrals In general, the error is “Error is order defines a convergence rate
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Zeroth-Order Approximation First-Order Approximation Second-Order Approximation ... Linear Systems What is a linear system? a system is linear if the system satisfies both the “addivity” and “homogeneity” properties Let be a mathematical operator (mapping), then is called linear if 1) “addivity” 2) “homogeneity” 1) and 2) are called the principle of superposition Example This is linear this is not linear **in order for a line to be linear, a line must pass through the origin **there is a distinction between a linear mapping and a “linear equation” The inverse of matrix A
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The inverse can be found as , where the adjoint is the transpose of the matrix of cofactors The Determinant is a scalar measure of a square matrix, where where Transpose - the transformation of rows to columns * If a matrix is symmetric
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