On Moore‐Penrose Pseudoinverse Computation for Stiffness Matrices Resulting from Higher Order Approximation

Abstract

Computing the pseudoinverse of a matrix is an essential component of many computational methods. It arises in statistics, graphics, robotics, numerical modeling, and many more areas. Therefore, it is desirable to select reliable algorithms that can perform this operation efficiently and robustly. A demanding benchmark test for the pseudoinverse computation was introduced. The stiffness matrices for higher order approximation turned out to be such tough problems and therefore can serve as good benchmarks for algorithms of the pseudoinverse computation. It was found out that only one algorithm, out of five known from literature, enabled us to obtain acceptable results for the pseudoinverse of the proposed benchmark test.

Design and Development of New Generation ALK Inhibitors Driven by Structure–Activity Relationship Studies of 2,4‐Diarylaminopyrimidine Derivatives

“…[30] In the implementation process, input weights and biases were randomly generated to compute the hidden layer output, which was then transformed using a sigmoid activation function to effectively capture complex nonlinear features. [31] Output weights were calculated using the Moore-Penrose pseudoinverse, [32] enabling the model to generate ELM predictions based on the hidden layer output. This approach provides a valuable supplement to the activity predictions of the Topomer CoMFA model.…”

Design and Development of New Generation ALK Inhibitors Driven by Structure–Activity Relationship Studies of 2,4‐Diarylaminopyrimidine Derivatives

“…[30] In the implementation process, input weights and biases were randomly generated to compute the hidden layer output, which was then transformed using a sigmoid activation function to effectively capture complex nonlinear features. [31] Output weights were calculated using the Moore-Penrose pseudoinverse, [32] enabling the model to generate ELM predictions based on the hidden layer output. This approach provides a valuable supplement to the activity predictions of the Topomer CoMFA model.…”

Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and $λ$-Effectiveness

“…3 See [15][16][17] for more recent application/analysis of the Moore-Penrose pseudoinverse. 4 Equivalently, X † X = P (ker X) ⊥ = P ran(X * ) 5 See e.g., [18] for the finite dimensional version.…”

“…Let µ be a singular Borel probability measure on [0, 1) with Fourier-Stieltjes transform µ. Let {α (λ) n } n≥0 be defined by (16) and let {h (λ) n } n≥0 be defined recursively by (17). Then, for all real λ ∈ (0, 2), and every n ≥ 0, we have…”

Impulse Delay in the Cardiac Conduction System

“…Therefore, each matrix here is presupposed Moore-Penrose invertible. Details on Moore-Penrose pseudoinverse of matrices may be found in [17,18] and in profuse literatures. The beauty of non-square systems is their less amenability to modelling errors [19].…”

“…consisting of triple matrices P, Q, R assumed to be positive definite, and where in the present case A 0 and A 1 are Moore-Penrose invertible [18,19,20]). Therefore:…”

Design and Development of New Generation ALK Inhibitors Driven by Structure–Activity Relationship Studies of 2,4‐Diarylaminopyrimidine Derivatives

“…[30] In the implementation process, input weights and biases were randomly generated to compute the hidden layer output, which was then transformed using a sigmoid activation function to effectively capture complex nonlinear features. [31] Output weights were calculated using the Moore-Penrose pseudoinverse, [32] enabling the model to generate ELM predictions based on the hidden layer output. This approach provides a valuable supplement to the activity predictions of the Topomer CoMFA model.…”

Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and $λ$-Effectiveness

“…3 See [15][16][17] for more recent application/analysis of the Moore-Penrose pseudoinverse. 4 Equivalently, X † X = P (ker X) ⊥ = P ran(X * ) 5 See e.g., [18] for the finite dimensional version.…”

“…Let µ be a singular Borel probability measure on [0, 1) with Fourier-Stieltjes transform µ. Let {α (λ) n } n≥0 be defined by (16) and let {h (λ) n } n≥0 be defined recursively by (17). Then, for all real λ ∈ (0, 2), and every n ≥ 0, we have…”

Impulse Delay in the Cardiac Conduction System

“…Therefore, each matrix here is presupposed Moore-Penrose invertible. Details on Moore-Penrose pseudoinverse of matrices may be found in [17,18] and in profuse literatures. The beauty of non-square systems is their less amenability to modelling errors [19].…”

“…consisting of triple matrices P, Q, R assumed to be positive definite, and where in the present case A 0 and A 1 are Moore-Penrose invertible [18,19,20]). Therefore:…”

Design and Development of New Generation ALK Inhibitors Driven by Structure–Activity Relationship Studies of 2,4‐Diarylaminopyrimidine Derivatives

“…[30] In the implementation process, input weights and biases were randomly generated to compute the hidden layer output, which was then transformed using a sigmoid activation function to effectively capture complex nonlinear features. [31] Output weights were calculated using the Moore-Penrose pseudoinverse, [32] enabling the model to generate ELM predictions based on the hidden layer output. This approach provides a valuable supplement to the activity predictions of the Topomer CoMFA model.…”

Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and $λ$-Effectiveness

“…3 See [15][16][17] for more recent application/analysis of the Moore-Penrose pseudoinverse. 4 Equivalently, X † X = P (ker X) ⊥ = P ran(X * ) 5 See e.g., [18] for the finite dimensional version.…”

“…Let µ be a singular Borel probability measure on [0, 1) with Fourier-Stieltjes transform µ. Let {α (λ) n } n≥0 be defined by (16) and let {h (λ) n } n≥0 be defined recursively by (17). Then, for all real λ ∈ (0, 2), and every n ≥ 0, we have…”

Impulse Delay in the Cardiac Conduction System

“…Therefore, each matrix here is presupposed Moore-Penrose invertible. Details on Moore-Penrose pseudoinverse of matrices may be found in [17,18] and in profuse literatures. The beauty of non-square systems is their less amenability to modelling errors [19].…”

“…consisting of triple matrices P, Q, R assumed to be positive definite, and where in the present case A 0 and A 1 are Moore-Penrose invertible [18,19,20]). Therefore:…”