Numerical Analysis Formula Cheat Sheet for Methods and Error

Numerical analysis formula cheat sheet provides methods for approximation, error calculation, interpolation, and integration, helping solve complex mathematical problems using numerical techniques.

Numerical Analysis Formula Cheat Sheet

Topic	Formula	Description
Absolute Error	|True − Approx|	Difference from exact value
Relative Error	|Error / True|	Accuracy measure
Percentage Error	(Error/True) × 100	Error in percent
Machine Epsilon	Smallest ε such that 1+ε ≠ 1	Precision limit
Condition Number	κ = |f′(x)|*	x/f(x)
Bisection Method	c = (a+b)/2	Root by halving interval
Convergence (Bisection)	Error ≤ (b−a)/2ⁿ	Accuracy bound
Fixed Point Iteration	xₙ₊₁ = g(xₙ)	Iterative method
Newton-Raphson	xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)	Fast root finding
Secant Method	xₙ₊₁ = xₙ − f(xₙ)(xₙ−xₙ₋₁)/(f(xₙ)−f(xₙ₋₁))	No derivative
Regula Falsi	Linear interpolation	Bracketing method
Order of Convergence	p	Speed of convergence
Interpolation	f(x) ≈ P(x)	Approximate function
Lagrange Polynomial	Σ yᵢLᵢ(x)	Interpolation formula
Newton Divided Diff	f[x₀,…,xₙ]	Recursive interpolation
Forward Difference	Δy = yₙ₊₁ − yₙ	Discrete difference
Backward Difference	∇y = yₙ − yₙ₋₁	Reverse difference
Central Difference	δy = (yₙ₊₁ − yₙ₋₁)/2	Symmetric difference
Trapezoidal Rule	∫ ≈ (h/2)(y₀ + yₙ + 2Σyᵢ)	Integration
Simpson’s 1/3	∫ ≈ (h/3)(y₀ + yₙ + 4Σodd + 2Σeven)	Better approx
Simpson’s 3/8	∫ ≈ (3h/8)(y₀ + yₙ + 3Σ + 2Σ)	Cubic approx
Gaussian Quadrature	Σ wᵢ f(xᵢ)	Weighted integration
Romberg Integration	Extrapolation of trapezoidal	High accuracy
Euler Method	yₙ₊₁ = yₙ + h f(xₙ,yₙ)	Basic ODE solver
Improved Euler	Avg slope method	Better Euler
Runge-Kutta (RK4)	Weighted slopes (k₁,k₂,k₃,k₄)	High accuracy ODE
Finite Difference	Replace derivatives with differences	PDE solving
FTCS Scheme	Forward time, centered space	Heat equation
Crank-Nicolson	Average of implicit & explicit	Stable scheme
LU Decomposition	A = LU	Solve linear systems
Gaussian Elimination	Row reduction	Solve equations
Jacobi Iteration	xᵢ = (bᵢ − Σaᵢⱼxⱼ)/aᵢᵢ	Iterative solver
Gauss-Seidel	Updated values used	Faster iteration
SOR Method	Relaxation factor ω	Accelerated method
Conjugate Gradient	Solve Ax=b efficiently	Large systems
Power Method	λ ≈ dominant eigenvalue	Eigenvalue method
Inverse Power	Smallest eigenvalue	Eigenvalue method
QR Algorithm	A = QR	Eigenvalue computation
FFT	O(n log n)	Fast transform
Stability	Small error growth	Reliable method
Stiff Equation	Requires stable methods	ODE difficulty

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Conclusion Numerical analysis is concerned with finding approximate solutions to mathematical problems by means of numerical methods. It is very common in engineering, science and computing. Numerical analysis aids in solving equations that are not solvable. It involves techniques such as interpolation, integration and root finding. Numbers: Computers are significant in numerical analysis. Studying this area enhances problem-solving and computation skills. It assists in error analysis, and enhancing accuracy. Simulations and modeling involve the use of numerical techniques. They play a crucial role in scientific research and engineering. All in all, numerical analysis is a valuable discipline that involves mathematics and computation. To conclude, numerical analysis helps solve complex problems using approximations. It is essential for scientific computing and real-world applications.