errors

errors - tional error oF ± M In N operations roundo²...

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IEEE Floating Point Area lnumber¯ r is represented on the computer by ¯ r r = ± (1 + f )2 n where the 1 is a phantom .Indoub lep rec i s ionfoa t ingpo in tona32 -b i two rd computer, there is one sign bit s ( s =0For+and s =1For - ), the mantissa 0 f< 1isallotted52bits(precision),andtheexponent - 1022 n 1023 is allotted the remaining 11 bits (range). r is stored as { s, f, x } ,wherethesh i Ftedexponent 1 x = n +1023 2046 = 2 11 - 2 x =0isreservedFor underfow ,and x =2 11 - 1correspondstoIn Fi F f =0 and to NaN iF f ² =0 . Check that: 1/0 = InF, 1/InF = 0, InF + InF = InF, 0/0 = NaN, InF - InF = NaN. Machine epsilon ± M is de±ned so that 1 + ± M is the next positive number aFter 1 on the computer. VeriFy that: ± M =2 - 52 2 . 22 × 10 - 16 realmax =(2 - ± M )2 1023 2 1024 1 . 8 × 10 308 realmin =2 - 1022 2 . 2 × 10 - 308 Either ± M or ±
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Unformatted text preview: tional error oF ± M . In N operations, roundo² errors may accumulate ran-domly (error ∼ √ N± M ), systematically (error ∼ N± M ), or catastrophically (1 / ( b-a ) may → ± InF). (2) Truncation or discretization error: An infnite series has to be approxi-mated by a fnite number oF terms on the computer. ±or example, a contin-uous real interval [ a, b ] is approximated by a fnite set oF gridpoints x = a , x 1 = x + Δ x , x 2 = x + 2 Δ x , . . . , x n = x + n Δ x = b ; or, as another example, e x ≈ N ± n =0 x n n ! , | x | < 1 (3) Numerical instability: roundo² error introduced early in the computation can be exponentially amplifed and swamp the true solution. ±or example, the Forward Euler method For the heat equation u t = u xx is stable only iF Δ t ≤ Δ x 2 / 2. ±orward Euler computed solutions to the heat equation with 20Δ x . 2...
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errors - tional error oF ± M In N operations roundo²...

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