The NAG library ‘machine’

Ford, Brian; Bentley, Jon Louis; Croz, J. J. Du; Hague, Stephen J.

doi:10.1002/spe.4380090107

Cited by 23 publications

(4 citation statements)

References 3 publications

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“…The fitting was carried out using a least squares minimization method algorithm from the Numerical Algorithms Group (NAG) (29, 30). The parameters were all constrained to be positive, as a negative value for any parameter would be physically unrealistic.…”

Section: Methodsmentioning

confidence: 92%

Quantitative magnetization transfer mapping of bound protons in multiple sclerosis

Tozer¹,

Ramani²,

Barker³

et al. 2003

Magnetic Resonance in Med

109

129

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Section: Methodsmentioning

confidence: 92%

Quantitative magnetization transfer mapping of bound protons in multiple sclerosis

Tozer¹,

Ramani²,

Barker³

et al. 2003

Magnetic Resonance in Med

109

129

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“…The problem has become one of nonlinear least squares and can be solved accurately with established algorithms [23,24]. Algorithms to solve nonlinear least squares require a starting estimate of b and in this case a simple linear fit to approximate efficiency curves of this chamber suffices.…”

Section: Least-squares Analysis Applied To Nonlinear Modelsmentioning

confidence: 99%

Ionization chamber efficiency curves

2007

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This paper describes the current status of ionization chamber efficiency curve modelling. Such curves, useful for ionization current prediction and for impurity corrections, cannot be obtained directly and are generally determined using empirical or Monte Carlo techniques. Each technique is described and practical examples are given. It is concluded that such models are a useful supplement to, rather than a replacement for, careful calibration and characterization measurements.

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“…(The NAG Library (see Ford et al, 1979) includes an algorithm based on this paper but using an extension of the structure-exploiting Givens scheme referred to above (Hayes, 1978)). …”

Section: The Minimum Norm Solutionmentioning

confidence: 99%

Practical spline approximation

Cox

1982

Topics in Numerical Analysis

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This two-part paper describes the use of polynomial spline functions for purposes of interpolation and approximation.The emphasis is on practical utility rather than detailed theory. Part I introduces polynomial splines, defines B-splines and treats the representation of splines in terms of B-splines. Part II deals with the statement and solution of spline interpolation and least squares spline approximation problems. It also discusses strategies for selecting particular solutions to spline approximation problems having nonunique solutions and techniques for automatic knot placement. ScopePolynomial spline functions (or simply polynomial splines or splines) have diverse application. They have been used to provide solutions to mathematical problems in interpolation, data and function approximation, ordinary and partial differential equations~ and integral equations. Splines have also been employed in many scientific and engineering applications; ones with which I personally have been concerned include instrument calibration, sonar signal analysis, highway visualization~ terrain following, computer aided design and manufacture, radioimmunoassay, telescope design and plant growth analysis.This paper places particular emphasis upon the algorithmic aspects of spline interpolation and least squares spline approximation. Additionally, the related tasks of evaluation, differentiation and indefinite integration of spline interpolants and approximants are discussed. Importance is attached to the use of a representation for splines of general order that affords a good balance between numerical stability and efficiency.Consideration is also given to the solution of the systems of linear algebraic equations that arise in the construction of spline interpolants and approximants.The matrices associated with these equations are banded in form, and the elimination and orthogonalization techniques described take full advantage of this structure.Splines are represented here in terms of a basis (i.e. as a linear combination of certain basis splines, just as polynomials can be expressed as a linear combination of certain basis polynomials such as Chebyshev or Legendre polynomials). Such a 80representation is used in preference to the redundant one consisting of a set of polynomial pieces together with continuity conditions at the joins. The truncated power functions form one possible basis, but the distinct advantages of employing instead a basis consisting of certain linear combinations of the truncated power functionsthe B-splines -will be demonstrated.References to equations take the form (e) or, if reference is to another part, p(e), where e is the equation number and p the part number.

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The NAG library ‘machine’

Cited by 23 publications

References 3 publications

Quantitative magnetization transfer mapping of bound protons in multiple sclerosis

Quantitative magnetization transfer mapping of bound protons in multiple sclerosis

Ionization chamber efficiency curves

Practical spline approximation

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