Advances in the Approximation of the Matrix Hyperbolic Tangent

Ibáñez, Javier; Alonso, Jesús José Beltrán de Heredia; Sastre, Jorge; Defez, Emilio; Alonso‐Jordá, Pedro

doi:10.3390/math9111219

Cited by 8 publications

(20 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also tested different variations of the formulas providing order m = 15+; however, there were no real solutions for any of them. The same also occurred in [44] (Section 2.2) with the hyperbolic tangent Taylor approximation of order m = 15+. In that case, we could find real solutions using an approximation of order m = 14+.…”

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 54%

“…where it is easy to show that the degree of y 22 (A) is 16. Therefore, we denote this approximation by S T 14+ (A), and similarly to [44] (Section 2.2), one obtains:…”

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 99%

“…In order to obtain all coefficients from these expressions, the nested procedure used in [43] (Sections 3.2 and 3.3) was applied. Analogous to [44] (p. 7), we employed the vpasolve function from the MATLAB Symbolic Computation Toolbox (https://es. mathworks.com/help/symbolic/vpasolve.html, accessed on 20 August 2021) with the random option to obtain different solutions of the nonlinear system arising for each order of approximation.…”

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 99%

See 2 more Smart Citations

An Improved Taylor Algorithm for Computing the Matrix Logarithm

et al. 2021

Self Cite

View full text Add to dashboard Cite

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

show abstract

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 54%

“…where it is easy to show that the degree of y 22 (A) is 16. Therefore, we denote this approximation by S T 14+ (A), and similarly to [44] (Section 2.2), one obtains:…”

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 99%

Section: Evaluation Of Higher-order Taylor-based Approximationsmentioning

confidence: 99%

See 1 more Smart Citation

An Improved Taylor Algorithm for Computing the Matrix Logarithm

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…If the use of new coe cients derived from those of 𝑝 is allowed, however, then even more economical schemes, such as [27, Algorithms B and C], are possible. Fuelled by the seminal work of Sastre [30], the topic has received renewed attention in recent years, and algorithms that combine this scheme with the scaling and squaring technique have been developed for the exponential [2,31], the sine and cosine [34,36], and the hyperbolic tangent [23].…”

Section: Degree-optimal Polynomialsmentioning

confidence: 99%

“…The formulae for the relative error in (22), (23), and (24) are exact, but they are also impractical:…”

Section: Round-o Errormentioning

confidence: 99%

Computational graphs for matrix functions

Jarlebring¹,

Fasi²,

Ringh³

2021

Preprint

View full text Add to dashboard Cite

Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective way to study existing techniques, improve them, and eventually derive new ones. As the accuracy of these matrix techniques is determined by the accuracy of their scalar counterparts, the design of algorithms for matrix functions can be viewed as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion akin to backpropagation. The Julia package GraphMatFun.jl offers the tools to generate and manipulate computational graphs, to optimize their coe cients, and to generate Julia, MATLAB, and C code to evaluate them e ciently. The software also provides the means to estimate the accuracy of an algorithm and thus obtain numerically reliable methods. For the matrix exponential, for example, using a particular form (degree-optimal) of polynomials produces algorithms that are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.

show abstract

Computational Graphs for Matrix Functions

Jarlebring

Fasi

Ringh

2022

ACM Trans. Math. Softw.

View full text Add to dashboard Cite

Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve them, and eventually derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion analogous to backpropagation. This paper describes GraphMatFun.jl , a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.

show abstract

Advances in the Approximation of the Matrix Hyperbolic Tangent

Cited by 8 publications

References 25 publications

An Improved Taylor Algorithm for Computing the Matrix Logarithm

An Improved Taylor Algorithm for Computing the Matrix Logarithm

Computational graphs for matrix functions

Computational Graphs for Matrix Functions

Contact Info

Product

Resources

About