Algorithm 741: least-squares solution of a linear, bordered, block-diagonal system of equations

1995 ◽  
Vol 21 (1) ◽  
pp. 20-25
Author(s):  
Richard D. Ray
Keyword(s):  
Least Squares ◽  
System Of Equations ◽  
Least Squares Solution ◽  
Diagonal System ◽  
Block Diagonal

The crystal and molecular structure of 1,1-bis(diphenylphosphino)-2,2-dimethylhydrazine

1976 ◽  
Vol 54 (5) ◽  
pp. 738-743 ◽  
Author(s):  
J. Brian Faught
Keyword(s):  
Molecular Structure ◽  
Least Squares ◽  
Crystal And Molecular Structure ◽  
Unit Cell ◽  
Space Group ◽  
Block Diagonal

The structure of 1,1-bis(diphenylphosphino)-2,2-dimethylhydrazine, [(C6H5)2P]2NN(CH3)2 has been determined crystallographically. The compound crystallizes from n-heptane in the space group P21/c with a = 8.910(1), b = 9.686(1), c = 27.489(4) Å, and β = 102.94(2)° with four molecules per unit cell. The structure was solved from 2669 independent reflections with I > 3σ(I) and refined by block diagonal least squares methods to R = 0.032 and Rω = 0.048. Each diphenylphosphino group is bonded to the same hydrazine nitrogen and the geometry about this nitrogen is nearly planar. The average dimensions of the structure are P—C = 1.828 ± 0.005, P—N = 1.715 ± 0.014, N—N = 1.451, and N—C = 1.457 ± 0.003 Å, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].

On the Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

2017 ◽  
Vol 9 (6) ◽  
Author(s):  
Leila Notash
Keyword(s):  
Least Squares ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Constrained Systems ◽  
Solution Set ◽  
Least Squares Solution ◽  
Interval Linear ◽  
Interval Linear Systems ◽  
Parameter Dependency ◽  
Interval Systems

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized, and implemented to identify the solution set, as well as the minimum 2-norm least-squares solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least-squares solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.

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