minimaxApprox {minimaxApprox}

Minimax Approximation of Functions

Description

Calculates minimax approximations to functions. Polynomial approximation uses the Remez (1962) algorithm. Rational approximation uses the Cody-Fraser-Hart (Cody et al., 1968) version of the algorithm. When using monomials as the polynomial basis, the Compensated Horner Scheme of Langlois et al. (2006) is used.

Usage

minimaxApprox(fn, lower, upper, degree, relErr = FALSE, basis ="Chebyshev",
              xi = NULL, opts = list())

Arguments

fn

function; A vectorized univariate function having x as its first argument. This could be a built-in R function, a predefined function, or an anonymous function defined in the call; see Examples.

lower

numeric; The lower bound of the approximation interval.

upper

numeric; The upper bound of the approximation interval.

degree

integer; Either a single value representing the requested degree for polynomial approximation or a vector of length 2 representing the requested degrees of the numerator and denominator for rational approximation.

relErr

logical; If TRUE, calculate the minimax approximation using relative error. The default is FALSE which uses absolute error.

basis

character; Which polynomial basis to use in the analysis. "Monomial" uses the standard x^k basis. "Chebyshev" uses the Chebyshev polynomials of the first kind, T_k. The default is "Chebyshev", and the parameter is case-insensitive and may be abbreviated.

xi

numeric; For rational approximation, a vector of initial points of the correct length—\sum(\code{degree}) + 2. If missing, the approximation will use the appropriate Chebyshev nodes. Polynomial approximation always uses Chebyshev nodes and will ignore xi with a message.

opts

list; Configuration options including:

• maxiter: integer; The maximum number of iterations to attempt convergence. Defaults to 100.

• miniter: integer; The minimum number of iterations before allowing convergence. Defaults to 10.

• conviter: integer; The number of successive iterations with the same results allowed before assuming no further convergence is possible. Defaults to 30. Will overwrite maxiter and miniter if conviter is explicitly passed and is larger than either one.

• showProgress: logical; If TRUE will print error values at each iteration.

• convrat: numeric; The convergence ratio tolerance. Defaults to 1 + 1 \times 10^{-9}. See Details.

• tol: numeric; The absolute difference tolerance. Defaults to 1 \times 10^{-14}. See Details.

• tailtol: numeric; The tolerance of the coefficient of the largest power of x to be ignored when performing the polynomial approximation a second time. Defaults to the smaller of 1 \times 10^{-10} or \frac{\code{upper} - \code{lower}}{10^6}. Set to NULL to skip the degree + 1 check completely. See Details.

• ztol: numeric; The tolerance for each polynomial or rational numerator or denominator coefficient's contribution to not to be set to 0. Similar to polynomial tailtol but applied at each step of the algorithm. Defaults to NULL which leaves all coefficients as they are regardless of magnitude. See Details.

Details

Convergence

The function implements the Remez algorithm using linear approximation, chiefly as described by Cody et al. (1968). Convergence is considered achieved when all three of the following criteria are met:

1. The observed error magnitudes are within tolerance of the expected error—the Distance Test.

2. The observed error magnitudes are within tolerance of each other—the Magnitude Test.

3. The observed error signs oscillate—the Oscillation Test.

“Within tolerance” can be met in one of two ways:

1. Difference: The difference between the absolute magnitudes is less than or equal to tol.

2. Ratio: The ratio between the absolute magnitudes of the larger and smaller is less than or equal to convrat.

For efficiency, the Distance Test is taken between the absolute value of the largest observed error and the absolute value of the expected error. Similarly, the Magnitude Test is taken between the absolute value of the largest observed error and the absolute value of the smallest observed error. Both tests can be passed by either being within tol or convrat as described above. However, when the Difference test returns values less than machine precision, it is ignored in favor of the Ratio test.

When the error values remain within tolerance of each other over conviter iterations, the algorithm will stop, as it is expected that no further precision will be gained by continued iterations.

Polynomial Evaluation

Monomial polynomials are evaluated using the Compensated Horner Scheme of Langlois et al. (2006) to enhance both stability and precision. Chebyshev polynomials are evaluated normally. There may be cases where the algorithm will fail using the monomial basis but succeed using Chebyshev polynomials and vice versa. The default is to use the Chebyshev polynomials.

Polynomial Algorithm “Singular Error” Response

When too high of a degree is requested for the tolerance of the algorithm, it often fails with a singular matrix error. In this case, for the polynomial version, the algorithm will try looking for an approximation of degree n + 1. If it finds one, and the contribution of that coefficient to the approximation is \le tailtol, it will ignore that coefficient and return the resulting degree n polynomial, as the largest coefficient is effectively 0. The contribution is measured by multiplying that coefficient by the endpoint with the larger absolute magnitude raised to the n + 1 power. This is done to prevent errors in cases where a very small coefficient is found on a range with very large absolute values and the resulting contribution to the approximation is not de minimis. Setting tailtol to NULL will skip the n + 1 test completely.

Close-to-Zero Tolerance

For each step of the algorithms' iterations, the contribution of the found coefficient to the total sum (as measured in the above section) is compared to the ztol option. When less than or equal to ztol, that coefficient is set to 0. Setting ztol to NULL skips the test completely. For intervals near or containing zero, setting this option to anything other than NULL may result in either non-convergence or poor results. It is recommended to keep it as NULL, although there are edge cases where it may allow convergence where a standard call may fail.

Value

minimaxApprox returns an object of class "minimaxApprox" which inherits from the class list.

The generic accessor function coef will extract the numerator and denominator vectors. There are also default print and plot methods.

An object of class "minimaxApprox" is a list containing the following components:

a

The polynomial or rational numerator coefficients. When using Chebyshev polynomials, these are the coefficients for T_k. When using monomials, these are the coefficients for x^k.

b

The rational denominator coefficients. When using Chebyshev polynomials, these are the coefficients for T_k. When using monomials, these are the coefficients for x^k. Missing for polynomial approximation.

aMono

When using Chebyshev polynomials, these are the polynomial or rational numerator coefficients for monomial expansion in x^k. Missing for monomial-based approximation.

bMono

When using Chebyshev polynomials, these are the rational denominator coefficients for monomial expansion in x^k. Missing for both polynomial and monomial-based rational approximation.

ExpErr

The absolute value of the expected error as calculated by the Remez algorithms.

ObsErr

The absolute value of largest observed error between the function and the approximation at the extremal points.

iterations

The number of iterations of the algorithm. This does not include any iterations required to converge the error value in rational approximation.

Extrema

The extrema at which the minimax error was achieved.

Warning

A logical flag indicating if any warnings were thrown.

The object also contains the following attributes:

type

"Rational" or "Polynomial".

basis

"Monomial" or "Chebyshev".

func

The function being approximated.

range

The range on which the function is being approximated.

relErr

A logical indicating that relative error was used. If FALSE, then absolute error was used.

tol

The tolerance used for the Distance Test.

convrat

The tolerance used for the Magnitude Test.

Note

At present, the algorithms are implemented using machine double precision, which means that the approximations are at best slightly worse. Research proceeds on more precise, stable, and efficient implementations. So long as the package remains in an experimental state—noted by a 0 major version—the API may change at any time.

Author(s)

Avraham Adler [e-mail]

References

Remez, E. I. (1962) General computational methods of Chebyshev approximation: The problems with linear real parameters. US Atomic Energy Commission, Division of Technical Information. AEC-tr-4491

Fraser W. and Hart J. F. (1962) “On the computation of rational approximations to continuous functions”, Communications of the ACM, 5(7), 401–403, doi:10.1145/368273.368578

Cody, W. J. and Fraser W. and Hart J. F. (1968) “Rational Chebyshev approximation using linear equations”, Numerische Mathematik, 12, 242–251, doi:10.1007/BF02162506

Langlois, P. and Graillat, S. and Louvet, N. (2006) “Compensated Horner Scheme”, in Algebraic and Numerical Algorithms and Computer-assisted Proofs. Dagstuhl Seminar Proceedings, 5391, doi:10.4230/DagSemProc.05391.3

See Also

minimaxEval, minimaxErr

Examples

minimaxApprox(exp, 0, 1, 5)                              # Built-in & polynomial

fn <- function(x) sin(x) ^ 2 + cosh(x)                   # Pre-defined
minimaxApprox(fn, 0, 1, c(2, 3), basis = "m")            # Rational

minimaxApprox(function(x) x ^ 3 / sin(x), 0.7, 1.6, 6L)  # Anonymous

fn <- function(x) besselJ(x, nu = 0)                     # More than one input
b0 <- 0.893576966279167522                               # Zero of besselY
minimaxApprox(fn, 0, b0, c(3L, 3L))                      # Cf. DLMF 3.11.19