About the Project

triangular matrices

AdvancedHelp

(0.002 seconds)

11—20 of 43 matching pages

11: 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 ) .

  • 12: 35.5 Bessel Functions of Matrix Argument
    35.5.5 𝟎 < 𝐗 < 𝐓 A ν 1 ( 𝐒 1 𝐗 ) | 𝐗 | ν 1 A ν 2 ( 𝐒 2 ( 𝐓 𝐗 ) ) | 𝐓 𝐗 | ν 2 d 𝐗 = | 𝐓 | ν 1 + ν 2 + 1 2 ( m + 1 ) A ν 1 + ν 2 + 1 2 ( m + 1 ) ( ( 𝐒 1 + 𝐒 2 ) 𝐓 ) , ν j , ( ν j ) > 1 , j = 1 , 2 ; 𝐒 1 , 𝐒 2 𝓢 ; 𝐓 𝛀 .
    13: 35.3 Multivariate Gamma and Beta Functions
    35.3.2 Γ m ( s 1 , , s m ) = 𝛀 etr ( 𝐗 ) | 𝐗 | s m 1 2 ( m + 1 ) j = 1 m 1 | ( 𝐗 ) j | s j s j + 1 d 𝐗 , s j , ( s j ) > 1 2 ( j 1 ) , j = 1 , , m .
    35.3.3 B m ( a , b ) = 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 2 ( m + 1 ) | 𝐈 𝐗 | b 1 2 ( m + 1 ) d 𝐗 , ( a ) , ( b ) > 1 2 ( m 1 ) .
    14: Bille C. Carlson
    In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. …
    15: 21.1 Special Notation
    g , h positive integers.
    g × h set of all g × h matrices with integer elements.
    Uppercase boldface letters are g × g real or complex matrices. …
    16: 35.2 Laplace Transform
    35.2.1 g ( 𝐙 ) = 𝛀 etr ( 𝐙 𝐗 ) f ( 𝐗 ) d 𝐗 ,
    35.2.3 f 1 f 2 ( 𝐓 ) = 𝟎 < 𝐗 < 𝐓 f 1 ( 𝐓 𝐗 ) f 2 ( 𝐗 ) d 𝐗 .
    17: 21.6 Products
    21.6.1 𝒦 = g × h 𝐓 / ( g × h 𝐓 g × h ) ,
    that is, 𝒦 is the set of all g × h matrices that are obtained by premultiplying 𝐓 by any g × h matrix with integer elements; two such matrices in 𝒦 are considered equivalent if their difference is a matrix with integer elements. …
    21.6.3 j = 1 h θ ( k = 1 h T j k 𝐳 k | 𝛀 ) = 1 𝒟 g 𝐀 𝒦 𝐁 𝒦 e 2 π i tr [ 1 2 𝐀 T 𝛀 𝐀 + 𝐀 T [ 𝐙 + 𝐁 ] ] j = 1 h θ ( 𝐳 j + 𝛀 𝐚 j + 𝐛 j | 𝛀 ) ,
    21.6.4 j = 1 h θ [ k = 1 h T j k 𝐜 k k = 1 h T j k 𝐝 k ] ( k = 1 h T j k 𝐳 k | 𝛀 ) = 1 𝒟 g 𝐀 𝒦 𝐁 𝒦 e 2 π i j = 1 h 𝐛 j 𝐜 j j = 1 h θ [ 𝐚 j + 𝐜 j 𝐛 j + 𝐝 j ] ( 𝐳 j | 𝛀 ) ,
    18: 35.4 Partitions and Zonal Polynomials
    19: 35.6 Confluent Hypergeometric Functions of Matrix Argument
    35.6.2 Ψ ( a ; b ; 𝐓 ) = 1 Γ m ( a ) 𝛀 etr ( 𝐓 𝐗 ) | 𝐗 | a 1 2 ( m + 1 ) | 𝐈 + 𝐗 | b a 1 2 ( m + 1 ) d 𝐗 , ( a ) > 1 2 ( m 1 ) , 𝐓 𝛀 .
    35.6.6 B m ( b 1 , b 2 ) | 𝐓 | b 1 + b 2 1 2 ( m + 1 ) F 1 1 ( a 1 + a 2 b 1 + b 2 ; 𝐓 ) = 𝟎 < 𝐗 < 𝐓 | 𝐗 | b 1 1 2 ( m + 1 ) F 1 1 ( a 1 b 1 ; 𝐗 ) | 𝐓 𝐗 | b 2 1 2 ( m + 1 ) F 1 1 ( a 2 b 2 ; 𝐓 𝐗 ) d 𝐗 , ( b 1 ) , ( b 2 ) > 1 2 ( m 1 ) .
    35.6.8 𝛀 | 𝐓 | c 1 2 ( m + 1 ) Ψ ( a ; b ; 𝐓 ) d 𝐓 = Γ m ( c ) Γ m ( a c ) Γ m ( c b + 1 2 ( m + 1 ) ) Γ m ( a ) Γ m ( a b + 1 2 ( m + 1 ) ) , ( a ) > ( c ) + 1 2 ( m 1 ) > m 1 , ( c b ) > 1 .
    20: 19.31 Probability Distributions
    More generally, let 𝐀 ( = [ a r , s ] ) and 𝐁 ( = [ b r , s ] ) be real positive-definite matrices with n rows and n columns, and let λ 1 , , λ n be the eigenvalues of 𝐀 𝐁 1 . …