About the Project

triangular decomposition

AdvancedHelp

(0.001 seconds)

1—10 of 14 matching pages

1: 13.27 Mathematical Applications
§13.27 Mathematical Applications
Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … …
2: 3.2 Linear Algebra
This yields a lower triangular matrix of the form …If we denote by 𝐔 the upper triangular matrix comprising the elements u j k in (3.2.3), then we have the factorization, or triangular decomposition, … We solve the system 𝐀 𝛿 𝐱 = 𝐫 for 𝛿 𝐱 , taking advantage of the existing triangular decomposition of 𝐀 to obtain an improved solution 𝐱 + 𝛿 𝐱 . … In the case that the orthogonality condition is replaced by 𝐒 -orthogonality, that is, 𝐯 j T 𝐒 𝐯 k = δ j , k , j , k = 1 , 2 , , n , for some positive definite matrix 𝐒 with Cholesky decomposition 𝐒 = 𝐋 T 𝐋 , then the details change as follows. …
3: 16.24 Physical Applications
The 3 j symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …
4: 26.13 Permutations: Cycle Notation
For the example (26.13.2), this decomposition is given by ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) = ( 1 , 3 ) ( 2 , 3 ) ( 2 , 5 ) ( 5 , 7 ) ( 6 , 8 ) . Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) = ( 2 , 3 ) ( 1 , 2 ) ( 4 , 5 ) ( 3 , 4 ) ( 2 , 3 ) ( 3 , 4 ) ( 4 , 5 ) ( 6 , 7 ) ( 5 , 6 ) ( 7 , 8 ) ( 6 , 7 ) : inv ( ( 1 , 3 , 2 , 5 , 7 ) ( 6 , 8 ) ) = 11 .
5: 9.19 Approximations
  • Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of Ai ( z ) , Ai ( z ) stored at the nodes. Ai ( z ) and Ai ( z ) are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of Ai ( z ) , Ai ( z ) at the node. Similarly for Bi ( z ) , Bi ( z ) .

  • 6: 19.14 Reduction of General Elliptic Integrals
    The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …
    7: 5.19 Mathematical Applications
    By decomposition into partial fractions (§1.2(iii)) …
    8: 1.3 Determinants, Linear Operators, and Spectral Expansions
    Determinants of Upper/Lower Triangular and Diagonal Matrices
    The determinant of an upper or lower triangular, or diagonal, square matrix 𝐀 is the product of the diagonal elements det ( 𝐀 ) = i = 1 n a i i . …
    9: Bibliography O
  • G. E. Ordóñez and D. J. Driebe (1996) Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.
  • 10: 1.2 Elementary Algebra
    𝐀 is an upper or lower triangular matrix if all a i j vanish for i > j or i < j , respectively. Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. …