{"id":22886,"date":"2023-03-01T09:32:45","date_gmt":"2023-03-01T09:32:45","guid":{"rendered":"https:\/\/www.booksofall.com\/cn\/?post_type=product&p=22886"},"modified":"2023-03-01T09:32:45","modified_gmt":"2023-03-01T09:32:45","slug":"tea-time-numerical-analysis-experiences-in-mathematics","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/cn\/tea-time-numerical-analysis-experiences-in-mathematics\/","title":{"rendered":"Tea Time Numerical Analysis – Experiences in Mathematics"},"content":{"rendered":"

Chapter 1 – Preliminaries<\/h2>\n

1.1 Accuracy Measuring Error\u00a0<\/b><\/p>\n

Numerical methods<\/a> are designed to approximate one thing or another. Sometimes roots, sometimes derivatives or definite integrals, or curves, or solutions of differential equations<\/a>. As numerical methods produce only approximations to these things, it is important to have some idea how accurate they are. Sometimes accuracy comes down to careful algebraic analysis<\/a>\u2014sometimes careful analysis of the calculus, and often careful analysis of Taylor polynomials<\/a>. But before we can tackle those details, we should discuss just how error and, therefore, accuracy are measured.<\/p>\n

There are two basic measurements of accuracy: absolute error and relative error<\/a>. Suppose that\u00a0p\u00a0<\/i>is the value we are approximating, and\u00a0p\u0303\u00a0<\/i>is an approximation of\u00a0p<\/i>. Then\u00a0p\u0303\u00a0<\/i>misses the mark by exactly the quantity\u00a0p\u0303\u2212 p<\/i>, the so-called error. Of course,\u00a0p\u0303\u00a0<\/i>\u2212\u00a0p\u00a0<\/i>will be negative when\u00a0p\u0303\u00a0<\/i>misses low. That is, when the approximation\u00a0p\u0303\u00a0<\/i>is less than the exact value\u00a0p<\/i>. On the other hand,\u00a0p\u0303\u00a0<\/i>\u2212\u00a0p\u00a0<\/i>will be positive when\u00a0p\u0303\u00a0<\/i>misses high. But generally, we are not concerned with whether our approximation is too high or too low. We just want to know how far off it is. Thus, we most often talk about the absolute error, |p\u0303 \u2212\u00a0p<\/i>|. You might recognize the expression |p\u0303 \u2212\u00a0p<\/i>| as the distance between\u00a0p\u0303\u00a0<\/i>and\u00a0p<\/i>, and that\u2019s not a bad way to think about absolute error.<\/p>\n

The absolute error in approximating\u00a0p\u00a0<\/i>=\u00a0\u03c0\u00a0<\/i>by the rational number<\/a> \"\". The absolute error in approximating\u00a0<\/span>\u03c0<\/i>5 by the rational number 16525\/<\/span>54 is | 16525 \/54 \u2212 \u03c0<\/i>5| \u2248 0.<\/i>00116. The absolute errors in these two approximations are nearly equal. To make the point more transparent,\u00a0\u03c0\u00a0<\/i>\u2248 3.<\/i>14159 and 22 7 \u2248 3.<\/i>14285, while \u03c0<\/i>5 \u2248 306.<\/i>01968 and 16525 54 \u2248 306.<\/i>01851. Each approximation begins to differ from its respective exact value in the thousandths place. And each is off by only 1 in the thousandths place.<\/p>\n

But there is something more going on here.\u00a0\u03c0\u00a0<\/i>is near 3 while\u00a0\u03c0<\/i>5 is near 300. To approximate\u00a0\u03c0\u00a0<\/i>accurate to the nearest one hundredth requires the approximation to agree with the exact value in only 3 place values\u2014the ones, tenths, and hundredths. To approximate\u00a0\u03c0<\/i>5 accurate to the nearest one hundredth requires the approximation to agree with the exact value in 5 place values\u2014the hundreds, tens, ones, tenths, and hundredths. To use more scientific language, we say that 22\/7 approximates\u00a0\u03c0\u00a0<\/i>accurate to 3 significant digits while 16525 54 approximates\u00a0\u03c0<\/i>5 accurate to 5 significant digits. Therein lies the essence of relative errors\u2014weighing the absolute error against the magnitude of the number being approximated. This is done by computing the ratio of the error to the exact value. Hence, the relative error in approximating\u00a0\u03c0\u00a0<\/i>by 22\/7 is \"\" while the relative error in approximating <\/span>\u03c0<\/i>5 by 16525\/54 is <\/span>\"\".<\/span> The relative errors differ by a factor of about 100 (equivalent to about two significant digits of accuracy) even though the absolute errors are nearly equal. In general, the relative error in approximating <\/span>p\u00a0<\/i>by\u00a0<\/span>p\u0303\u00a0<\/i>is given by\u00a0<\/span>\"\".<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"