scholarly journals Rounding Error Analysis of Finite Settling Time Control with Floating Point Arithmetic

1984 ◽  
Vol 20 (1) ◽  
pp. 36-42
Author(s):  
Keiji HASHIMOTO ◽  
Hidenori KIMURA
Keyword(s):  
Error Analysis ◽  
Settling Time ◽  
Floating Point ◽  
Rounding Error ◽  
Rounding Error Analysis ◽  
Floating Point Arithmetic ◽  
Time Control ◽  
Point Arithmetic

scholarly journals Accurately computing the log-sum-exp and softmax functions

2020 ◽  
Author(s):  
Pierre Blanchard ◽  
Desmond J Higham ◽  
Nicholas J Higham
Keyword(s):  
Numerical Experiments ◽  
Data Science ◽  
Floating Point ◽  
Condition Numbers ◽  
Rounding Error ◽  
Floating Point Arithmetic ◽  
Error Analyses ◽  
Software Implementations ◽  
Similar Accuracy ◽  
Point Arithmetic

Abstract Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low-precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic and shifting can introduce subtractive cancellation. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy.

Evaluation of Legendre polynomials by a three-term recurrence in floating-point arithmetic

2018 ◽  
Vol 40 (1) ◽  
pp. 587-605
Author(s):  
Tomasz Hrycak ◽  
Sebastian Schmutzhard
Keyword(s):  
Error Analysis ◽  
Error Estimates ◽  
Legendre Polynomials ◽  
Cross Product ◽  
Floating Point ◽  
Legendre Functions ◽  
Floating Point Arithmetic ◽  
The Cross ◽  
Point Arithmetic

Abstract We prove new error estimates for a three-term recurrence that is used to compute Legendre polynomials. To this end we derive a bilinear representation of the cross-product of Legendre functions and demonstrate estimates of the cross-product that are needed in the error analysis of the recurrence.

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