Differential and linear approximation

What is it?

Formulas

$$f(a + h) = f(a) + f'(a)\,h + o(h)$$

$$\frac{|\Delta f|}{|f|} \approx \left|\frac{x f'(x)}{f(x)}\right|\frac{|\Delta x|}{|x|}$$

relative error amplification (condition number)

Where it shows up in computing

• Scientific computing★★★★★frequentScientific computing and algorithms

  First-order error propagation estimates how measurement and rounding errors affect results.

• Control theory★★★★★frequentRobotics and control

  Nonlinear plants are linearized around an operating point and controlled with linear theory.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Conditioning
• Newton's method
• Taylor polynomial
• Total differential and linearization
• Euler's method

Exercises