Alright, future engineers!
**Fixed-Point Iteration:** Finds roots by rearranging `f(x)=0` to `x=g(x)` then repeating `x_new = g(x_old)`.
Ex: For `x = cos(x)`, set `x_new = cos(x_old)`.
Pro-Tip: For convergence, `|g'(x)| < 1` near the root is key!
#NumericalMethods #Iteration #STEM #StudyNotes

Alright, future engineers!
**Euler's Method:** Numerically approximates solutions to first-order ODEs step-by-step using tangent lines.
Ex: `y_n+1 = y_n + h * f(x_n, y_n)`
Pro-Tip: Smaller step size (`h`) improves accuracy but increases computation!
#ODEs #NumericalMethods #STEM #StudyNotes

**Truncation Error:** Error from approximating an infinite mathematical process (like a series or integral) with a finite one.
Ex: `e^x ≈ 1+x` incurs truncation error.
Pro-Tip: It's systematic! Reduce it by taking more terms or using smaller step sizes.
#ErrorAnalysis #NumericalMethods #STEM #StudyNotes

Alright, future engineers!
**Newton-Raphson Method:** Iteratively finds a function's root (where f(x)=0).
Formula: `x_n+1 = x_n - f(x_n)/f'(x_n)`
Pro-Tip: A good initial guess is crucial for rapid convergence!
#RootFinding #NumericalMethods #STEM #StudyNotes

Simpson's Rule: Approximates definite integrals using parabolic segments for greater accuracy than trapezoids. Formula (1/3 rule for 2 segments): `(h/3) * (y0 + 4y1 + y2)`. Pro-Tip: Needs an even number of subintervals for classic application!
#NumericalMethods #Integration #STEM #StudyNotes

Alright, future engineers!
**Trapezoidal Rule:** Approximates definite integrals by summing the areas of trapezoids under a curve.
Think: `(h/2) * (y_start + y_end)` for each segment.
Pro-Tip: Smaller segment widths ('h') increase accuracy, but also computation time!
#NumericalMethods #Calculus #STEM #StudyNotes

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**Error Bound:** Max possible error for a numerical approximation.
Ex: `|True - Approx| <= E_b`.
Pro-Tip: Crucial for knowing if your solution meets required precision for engineering tasks!
#NumericalMethods #Accuracy #STEM #StudyNotes

Alright, future engineers!
**Euler's Method:** Approximates ODE solutions by taking tiny linear steps from an initial point.
Ex: `y_n+1 = y_n + h * f(x_n, y_n)`
Pro-Tip: Smaller 'h' (step size) gives better accuracy, but requires more computations!
#ODEs #NumericalMethods #STEM #StudyNotes

Error bounds quantify the max possible error in a numerical approximation. Ex: For Trapezoidal Rule, Error <= `(b-a)h^2/12 * max|f''(x)|`. Pro-Tip: Essential for knowing if your approximation is good enough for design specs! #NumericalMethods #ErrorAnalysis #STEM #StudyNotes

Alright, future engineers!
**Rounding Error:** Error from representing exact numbers with finite digits.
Ex: 1/3 is 0.333... but stored as 0.3333.
Pro-Tip: It accumulates! Mind precision in long calculations, esp. with huge/tiny numbers.
#NumericalMethods #ErrorAnalysis #STEM #StudyNotes