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Numerical certification

From Wikipedia, the free encyclopedia

Numerical certification is the process of verifying the correctness of a candidate solution to a system of equations. In (numerical) computational mathematics, such as numerical algebraic geometry, candidate solutions are computed algorithmically, but there is the possibility that errors have corrupted the candidates. For instance, in addition to the inexactness of input data and candidate solutions, numerical errors or errors in the discretization of the problem may result in corrupted candidate solutions. The goal of numerical certification is to provide a certificate which proves which of these candidates are, indeed, approximate solutions.

Methods for certification can be divided into two flavors: a priori certification and a posteriori certification. A posteriori certification confirms the correctness of the final answers (regardless of how they are generated), while a priori certification confirms the correctness of each step of a specific computation. A typical example of a posteriori certification is Smale's alpha theory, while a typical example of a priori certification is interval arithmetic.

Certificates

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A certificate for a root is a computational proof of the correctness of a candidate solution. For instance, a certificate may consist of an approximate solution , a region containing , and a proof that contains exactly one solution to the system of equations.

In this context, an a priori numerical certificate is a certificate in the sense of correctness in computer science. On the other hand, an a posteriori numerical certificate operates only on solutions, regardless of how they are computed. Hence, a posteriori certification is different from algorithmic correctness – for an extreme example, an algorithm could randomly generate candidates and attempt to certify them as approximate roots using a posteriori certification.

A posteriori certification methods

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There are a variety of methods for a posteriori certification, including

Alpha theory

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The cornerstone of Smale's alpha theory is bounding the error for Newton's method. Smale's 1986 work[1] introduced the quantity , which quantifies the convergence of Newton's method. More precisely, let be a system of analytic functions in the variables , the derivative operator, and the Newton operator. The quantities and are used to certify a candidate solution. In particular, if then is an approximate solution for , i.e., the candidate is in the domain of quadratic convergence for Newton's method. In other words, if this inequality holds, then there is a root of so that iterates of the Newton operator converge as

The software package alphaCertified provides an implementation of the alpha test for polynomials by estimating and .[2]

Interval Newton and Krawczyck methods

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Suppose is a function whose fixed points correspond to the roots of . For example, the Newton operator has this property. Suppose that is a region, then,

  1. If maps into itself, i.e., , then by Brouwer fixed-point theorem, has at least one fixed point in , and, hence has at least one root in .
  2. If is contractive in a region containing , then there is at most one root in .

There are versions of the following methods over the complex numbers, but both the interval arithmetic and conditions must be adjusted to reflect this case.

Interval Newton method

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In the univariate case, Newton's method can be directly generalized to certify a root over an interval. For an interval , let be the midpoint of . Then, the interval Newton operator applied to is

In practice, any interval containing can be used in this computation. If is a root of , then by the mean value theorem, there is some such that . In other words, . Since contains the inverse of at all points of , it follows that . Therefore, .

Furthermore, if , then either is a root of and or . Therefore, is at most half the width of . Therefore, if there is some root of in , the iterative procedure of replacing by will converge to this root. If, on the other hand, there is no root of in , this iterative procedure will eventually produce an empty interval, a witness to the nonexistence of roots.

See interval Newton method for higher dimensional analogues of this approach.

Krawczyck method

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Let be any invertible matrix in . Typically, one takes to be an approximation to . Then, define the function We observe that is a fixed of if and only if is a root of . Therefore the approach above can be used to identify roots of . This approach is similar to a multivariate version of Newton's method, replacing the derivative with the fixed matrix .

We observe that if is a compact and convex region and , then, for any , there exist such that

Let be the Jacobian matrix of evaluated on . In other words, the entry consists of the image of over . It then follows that where the matrix-vector product is computed using interval arithmetic. Then, allowing to vary in , it follows that the image of on satisfies the following containment: where the calculations are, once again, computed using interval arithmetic. Combining this with the formula for , the result is the Krawczyck operator

where is the identity matrix.

If , then has a fixed point in , i.e., has a root in . On the other hand, if the maximum matrix norm using the supremum norm for vectors of all matrices in is less than , then is contractive within , so has a unique fixed point.

A simpler test, when is an axis-aligned parallelepiped, uses , i.e., the midpoint of . In this case, there is a unique root of if

where is the length of the longest side of .

Miranda test

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  • Miranda test (Yap, Vegter, Sharma)

A priori certification methods

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  • Interval arithmetic (Moore, Arb, Mezzarobba)
  • Condition numbers (Beltran–Leykin)

Interval arithmetic

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Interval arithmetic can be used to provide an a priori numerical certificate by computing intervals containing unique solutions. By using intervals instead of plain numeric types during path tracking, resulting candidates are represented by intervals. The candidate solution-interval is itself the certificate, in the sense that the solution is guaranteed to be inside the interval.

Condition numbers

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Numerical algebraic geometry solves polynomial systems using homotopy continuation and path tracking methods. By monitoring the condition number for a tracked homotopy at every step, and ensuring that no two solution paths ever intersect, one can compute a numerical certificate along with a solution. This scheme is called a priori path tracking.[3]

Non-certified numerical path tracking relies on heuristic methods for controlling time step size and precision.[4] In contrast, a priori certified path tracking goes beyond heuristics to provide step size control that guarantees that for every step along the path, the current point is within the domain of quadratic convergence for the current path.

References

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  1. ^ Smale, Steve (1986). "Newton's method estimates from data at one point". The merging of disciplines: new directions in pure, applied, and computational mathematics: 185–196.
  2. ^ Hauenstein, Jonathan; Sottile, Frank (2012). "Algorithm 921: alphaCertified: certifying solutions to polynomial systems". ACM Transactions on Mathematical Software. 38 (4): 28. doi:10.1145/2331130.2331136.
  3. ^ Beltran, Carlos; Leykin, Anton (2012). "Certified numerical homotopy tracking". Experimental Mathematics. 21 (1): 69–83.
  4. ^ Bates, Daniel; Hauenstein, Jonathan; Sommese, Andrew; Wampler, Charles (2009). "Stepsize control for path tracking". Contemporary Mathematics. 496 (21).