Tuesday, Aug 26, 2025
numerical analysis = mathematical analysis of numerical methods for solving problems in science and engineering
We need to learn about the methods before we can analyze them
sneak preview of Ch 2: Newton's method for finding a root of a function (successive linearization)
imlementation of Newton's method in Python to solve 8x = e − x
Python 1: import modules, define and calling functions, iteration, exponentiation operator
analysis concerns: convergence guaranteed when? converges to root? how fast is convergence?
machine numbers, fixed amount of memory (e.g. 8 bytes = 64 bits)
number of numbers, uniform spacing?
there exists a largest, a smallest greater than 1
"scientific notation": allows very large and very small, and yields approximately uniform relative spacing
round-off error
place-value notation in base b
binary representation of real numbers, conversion of decimal integers and fractions to binary, examples
IEE754 "double" 64-bit: sign bit, d=11-bit exponent biased by 1023, m=52-bit mantissa
binary-hexadecimal conversion
some special numbers and their codes: 1, -1, smallest > 1, hexcodein.c, hexcodeout.c