#nummethods — Public Fediverse posts
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**Secant Method:** Finds `f(x)=0` roots via secant line using two prior points.
Ex: `x_n+1 = x_n - f(x_n) * (x_n - x_n-1) / (f(x_n) - f(x_n-1))`
Pro-Tip: No derivative needed! Faster than Bisection; requires two initial guesses.
#NumMethods #RootFinding #STEM #StudyNotes -
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**Fixed-Point Iteration:** Finds `x` where `x = g(x)` by successively applying `g` to the previous guess.
Ex: `x_n+1 = g(x_n)`.
Pro-Tip: For convergence, ensure `|g'(x)| < 1` near the root! Choose `g(x)` carefully.
#NumMethods #Iteration #STEM #StudyNotes -
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**Euler's Method:** Approximates ODE solutions by taking small linear steps.
Ex: `y_n+1 = y_n + h * f(x_n, y_n)`
Pro-Tip: Smaller `h` improves accuracy but cranks up computation! Balance for efficiency.
#NumMethods #ODEs #STEM #StudyNotes -
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**Bisection Method:** Finds `f(x)=0` roots by halving intervals `[a,b]` where `f(a)` and `f(b)` have opposite signs.
Ex: `c = (a+b)/2`; update `[a,c]` or `[c,b]`.
Pro-Tip: Guaranteed to converge, but it's slower than others!
#NumMethods #RootFinding #STEM #StudyNotes -
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**Numerical Integration:** Approximates definite integrals (area under curve) when exact solutions are tough.
Ex: Trapezoidal Rule for `[a,b]` is `h/2 * (f(a) + f(b))` for one segment.
Pro-Tip: Smaller step size `h` = greater accuracy (but more computation)!
#NumMethods #Calculus #STEM #StudyNotes -
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**Error Bounds:** Max possible error in numerical results; they gauge reliability!
Ex: Simpson's Rule: Truncation error is O(h^4).
Pro-Tip: Smaller 'h' for better methods can still mean *worse* accuracy due to round-off!
#NumMethods #Accuracy #STEM #StudyNotes -
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**Round-off Error:** Error from storing real numbers with finite precision in a computer.
Ex: `1/3` stored as `0.3333333` loses precision.
Pro-Tip: Accumulates! Small errors can become large in long calc chains.
#NumMethods #ErrorAnalysis #STEM #StudyNotes -
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**Newton-Raphson:** Finds f(x)=0 roots by iteratively approximating them with tangent lines.
Ex: `x_n+1 = x_n - f(x_n)/f'(x_n)`
Pro-Tip: Super fast, but needs a good initial guess and the derivative f'(x)!
#NumMethods #RootFinding #STEM #StudyNotes -
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**Error Bound:** Max diff between numerical approx & true value.
Ex: Reveals worst-case inaccuracy of your solution.
Pro-Tip: Don't just solve, know its limits! Crucial for reliability.
#NumMethods #ErrorAnalysis #STEM #StudyNotes -
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**Bisection Method:** Finds f(x)=0 roots by repeatedly halving an interval where f(x) changes sign.
Ex: If f(a)<0 & f(b)>0, root is in [a,b]. Midpoint splits it.
Pro-Tip: Guarantees convergence if you bracket a root!
#NumMethods #RootFinding #STEM #StudyNotes -
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**Euler's Method:** A first-order numerical procedure for solving initial value problems of ODEs.
Ex: `y_n+1 = y_n + h * f(t_n, y_n)`
Pro-Tip: Simple, but error increases with larger step sizes (h)!
#NumMethods #ODEs #STEM #StudyNotes -
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**Bisection Method:** Systematically finds a root by repeatedly halving an interval where a sign change indicates a root's presence.
Ex: If `f(a)<0` & `f(b)>0`, root is in `[a,b]`. New interval is `[a, (a+b)/2]` or `[(a+b)/2, b]`.
Pro-Tip: It ALWAYS converges (given a sign change!), but it's often slower than other methods.
#NumMethods #RootFinding #STEM #StudyNotes -
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**Relative Error:** Error normalized by the true value, showing accuracy proportionally.
Ex: True=10, Approx=9.8 => E_rel = |(10-9.8)/10| = 0.02 (2%)
Pro-Tip: Essential for comparing accuracy when scales differ vastly! -
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**Newton-Raphson:** Iteratively finds roots by using tangent lines to refine guesses.
Ex: `x_new = x_old - f(x_old) / f'(x_old)`.
Pro-Tip: A good initial guess is CRUCIAL for convergence. Bad ones can lead to divergence!
#NumMethods #Calculus #STEM #StudyNotes -
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**Simpson's Rule:** Numerical integration using parabolic arcs for higher accuracy.
Ex: Area `(h/3) * (f(a) + 4f(a+h) + f(a+2h))` for one segment.
Pro-Tip: More accurate than Trapezoidal, but needs an even # of subintervals!
#NumMethods #Integration #STEM #StudyNotes -
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**Truncation Error:** Error from approximating infinite math processes with finite ones.
Ex: Using finite terms of a Taylor series.
Pro-Tip: It's *method-dependent*! Smaller step size often helps, but beware round-off error.
#NumMethods #ErrorAnalysis #STEM #StudyNotes -
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**Trapezoidal Rule:** Approximates definite integrals by summing areas of trapezoids under the curve.
Ex: Area of one segment `0.5 * h * (f(a) + f(b))`.
Pro-Tip: Smaller step size 'h' dramatically boosts accuracy! More segments = better estimate.
#NumMethods #Calculus #STEM #StudyNotes -
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**Newton-Raphson:** Iteratively finds roots of f(x)=0 by approximating with tangent lines.
Ex: `x_n+1 = x_n - f(x_n)/f'(x_n)`
Pro-Tip: A good initial guess is crucial for fast convergence and to avoid divergence!
#NumMethods #RootFinding #STEM #StudyNotes -
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**Truncation Error:** Error from stopping an infinite process (like a series) prematurely.
Ex: Using `1+x` for `e^x` instead of `1+x+x^2/2!` introduces truncation error.
Pro-Tip: Smaller step sizes or more terms reduce it!
#ErrorAnalysis #NumMethods #STEM #StudyNotes -
Alright, future engineers!
**Truncation Error:** Error from stopping an infinite process (like a series) prematurely.
Ex: Using `1+x` for `e^x` instead of `1+x+x^2/2!` introduces truncation error.
Pro-Tip: Smaller step sizes or more terms reduce it!
#ErrorAnalysis #NumMethods #STEM #StudyNotes -
**Bisection Method:** Finds `f(x)=0` roots by repeatedly halving an interval with a sign change.
Ex: If `f(a)f(b) < 0`, root in `[a,b]`. New `c=(a+b)/2`.
Pro-Tip: Slow but *guaranteed* convergence if a root is bracketed. Reliability first!
#NumMethods #RootFinding #STEM #StudyNotes -
Alright, future engineers!
**Euler's Method:** Approximates solutions to ODEs (dy/dx=f(x,y)) by stepping along tangent lines.
Ex: `y_n+1 = y_n + h * f(x_n, y_n)`
Pro-Tip: Smaller step size 'h' boosts accuracy but adds steps! Balance is key.
#NumMethods #ODEs #STEM #StudyNotes -
Alright, future engineers!
**Newton-Raphson:** Iteratively finds roots of f(x)=0 by following tangent lines to approximate the zero.
Ex: `x_n+1 = x_n - f(x_n)/f'(x_n)`
Pro-Tip: A good initial guess means faster, surer convergence!
#NumMethods #RootFinding #STEM #StudyNotes -
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**Trapezoidal Rule:** Approximates `∫f(x)dx` by summing trapezoid areas under the curve.
Ex: `h/2 * (f(a)+f(b))` for `[a,b]`, where `h=b-a`.
Pro-Tip: Smaller `h` (more trapezoids) usually means better accuracy!
#NumMethods #Integration #STEM #StudyNotes -
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**Round-off Error:** Error from finite precision in computer arithmetic.
Ex: `1/3` stored as `0.333...3` isn't exact.
Pro-Tip: Accumulates! Can dominate when step sizes are tiny, offsetting truncation error gains.
#NumMethods #ErrorAnalysis #STEM #StudyNotes -
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**Truncation Error:** The error from stopping an infinite calculation early (finite approximation).
Ex: `e^x approx 1+x` (first two Taylor terms).
Pro-Tip: Smaller steps reduce it, but can increase *round-off* error. It's a trade-off!
#NumMethods #ErrorAnalysis #STEM #StudyNotes -
Alright, future engineers!
**Newton-Raphson:** Iteratively finds function roots by following tangent lines to the x-axis.
Ex: `x_next = x_curr - f(x_curr) / f'(x_curr)`
Pro-Tip: Requires a good initial guess! A near-zero `f'(x)` can cause divergence.
#NumMethods #RootFinding #STEM #StudyNotes -
Alright, future engineers!
**Euler's Method:** Approximates ODE solutions by stepping along tangent lines from an initial condition.
Ex: `y_next = y_curr + h * f(t_curr, y_curr)`
Pro-Tip: Smaller step size `h` usually improves accuracy, but increases computation!
#NumMethods #ODEs #STEM #StudyNotes -
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**Newton-Raphson:** Iteratively approximates a function's root using tangents.
Ex: `x_next = x - f(x) / f'(x)`
Pro-Tip: Your initial guess matters! A poor start can lead to divergence.
#NumMethods #EngineeringMath #STEM #StudyNotes -
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**Trapezoidal Rule:** Approximates definite integrals by summing areas of trapezoids under the curve.
Ex: `h/2 * (f(x0) + f(x1))` for one interval.
Pro-Tip: More subintervals (smaller `h`) generally means better accuracy! -
Alright, future engineers!
**Trapezoidal Rule:** Approximates definite integrals by summing areas of trapezoids under the curve.
Ex: `h/2 * (f(x0) + f(x1))` for one interval.
Pro-Tip: More subintervals (smaller `h`) generally means better accuracy!