List of numerical analysis topics

This is a list of numerical analysis topics. == General == Validated numerics Iterative method Rate of convergence — the speed at which a convergent sequence approaches its limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution Series acceleration — methods to accelerate the speed of convergence of a series Aitken's delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums Van Wijngaarden transformation — for accelerating the convergence of an alternating series Abramowitz and Stegun — book containing formulas and tables of many special functions Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun Curse of dimensionality Local convergence and global convergence — whether you need a good initial guess to get convergence Superconvergence Discretization Difference quotient Complexity: Computational complexity of mathematical operations Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs Symbolic-numeric computation — combination of symbolic and numeric methods Cultural and historical aspects: History of numerical solution of differential equations using computers Hundred-dollar, Hundred-digit Challenge problems — list of ten problems proposed by Nick Trefethen in 2002 Timeline of numerical analysis after 1945 General classes of methods: Collocation method — discretizes a continuous equation by requiring it only to hold at certain points Level-set method Level set (data structures) — data structures for representing level sets Sinc numerical methods — methods based on the sinc function, sinc(x) = sin(x) / x ABS methods == Error == Error analysis (mathematics) Approximation Approximation error Catastrophic cancellation Condition number Discretization error Floating point number Guard digit — extra precision introduced during a computation to reduce round-off error Truncation — rounding a floating-point number by discarding all digits after a certain digit Round-off error Numeric precision in Microsoft Excel Arbitrary-precision arithmetic Interval arithmetic — represent every number by two floating-point numbers guaranteed to have the unknown number between them Interval contractor — maps interval to subinterval which still contains the unknown exact answer Interval propagation — contracting interval domains without removing any value consistent with the constraints See also: Interval boundary element method, Interval finite element Loss of significance Numerical error Numerical stability Error propagation: Propagation of uncertainty Residual (numerical analysis) Relative change and difference — the relative difference between x and y is |x − y| / max(|x|, |y|) Significant figures Artificial precision — when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input False precision — giving more significant figures than appropriate Sterbenz lemma Truncation error — error committed by doing only a finite numbers of steps Well-posed problem Affine arithmetic == Elementary and special functions == Unrestricted algorithm Summation: Kahan summation algorithm Pairwise summation — slightly worse than Kahan summation but cheaper Binary splitting 2Sum Multiplication: Multiplication algorithm — general discussion, simple methods Karatsuba algorithm — the first algorithm which is faster than straightforward multiplication Toom–Cook multiplication — generalization of Karatsuba multiplication Schönhage–Strassen algorithm — based on Fourier transform, asymptotically very fast Fürer's algorithm — asymptotically slightly faster than Schönhage–Strassen Division algorithm — for computing quotient and/or remainder of two numbers Long division Restoring division Non-restoring division SRT division Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q. Goldschmidt division Exponentiation: Exponentiation by squaring Addition-chain exponentiation Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal).

Source: Wikipedia — List of numerical analysis topics (CC BY-SA 4.0)

List of numerical analysis topics

This is a list of numerical analysis topics. == General == Validated numerics Iterative method Rate of convergence — the speed at which a convergent sequence approaches its limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution Series acceleration — methods to accelerate the speed of convergence of a series Aitken's delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums Van Wijngaarden transformation — for accelerating the convergence of an alternating series Abramowitz and Stegun — book containing formulas and tables of many special functions Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun Curse of dimensionality Local convergence and global convergence — whether you need a good initial guess to get convergence Superconvergence Discretization Difference quotient Complexity: Computational complexity of mathematical operations Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs Symbolic-numeric computation — combination of symbolic and numeric methods Cultural and historical aspects: History of numerical solution of differential equations using computers Hundred-dollar, Hundred-digit Challenge problems — list of ten problems proposed by Nick Trefethen in 2002 Timeline of numerical analysis after 1945 General classes of methods: Collocation method — discretizes a continuous equation by requiring it only to hold at certain points Level-set method Level set (data structures) — data structures for representing level sets Sinc numerical methods — methods based on the sinc function, sinc(x) = sin(x) / x ABS methods == Error == Error analysis (mathematics) Approximation Approximation error Catastrophic cancellation Condition number Discretization error Floating point number Guard digit — extra precision introduced during a computation to reduce round-off error Truncation — rounding a floating-point number by discarding all digits after a certain digit Round-off error Numeric precision in Microsoft Excel Arbitrary-precision arithmetic Interval arithmetic — represent every number by two floating-point numbers guaranteed to have the unknown number between them Interval contractor — maps interval to subinterval which still contains the unknown exact answer Interval propagation — contracting interval domains without removing any value consistent with the constraints See also: Interval boundary element method, Interval finite element Loss of significance Numerical error Numerical stability Error propagation: Propagation of uncertainty Residual (numerical analysis) Relative change and difference — the relative difference between x and y is |x − y| / max(|x|, |y|) Significant figures Artificial precision — when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input False precision — giving more significant figures than appropriate Sterbenz lemma Truncation error — error committed by doing only a finite numbers of steps Well-posed problem Affine arithmetic == Elementary and special functions == Unrestricted algorithm Summation: Kahan summation algorithm Pairwise summation — slightly worse than Kahan summation but cheaper Binary splitting 2Sum Multiplication: Multiplication algorithm — general discussion, simple methods Karatsuba algorithm — the first algorithm which is faster than straightforward multiplication Toom–Cook multiplication — generalization of Karatsuba multiplication Schönhage–Strassen algorithm — based on Fourier transform, asymptotically very fast Fürer's algorithm — asymptotically slightly faster than Schönhage–Strassen Division algorithm — for computing quotient and/or remainder of two numbers Long division Restoring division Non-restoring division SRT division Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q. Goldschmidt division Exponentiation: Exponentiation by squaring Addition-chain exponentiation Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal).

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