Ddisna Dopgtr Dopmtr Dorgtr Dormtr Dpteqr Dsbtrd Dsptrd Dstebz Dstedc Dstein Dstemr Dsteqr Dsterf Dsytrd

Dsytrd() Sub Dsytrd ( Uplo As  String, N As  Long, A() As  Double, D() As  Double, E() As  Double, Tau() As  Double, Info As  Long  ) Reduces a real symmetric matrix to tridiagonal form Purpose This routine reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q^T * A * Q = T. Parameters [in] Uplo = "U": Upper triangle of A is stored. = "L": Lower triangle of A is stored. [in] N Order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= N, LA2 >= N) [in] The symmetric matrix A. If Uplo = "U", the leading N x N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If Uplo = "L", the leading N x N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. [out] If Uplo = "U", the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array Tau(), represent the orthogonal matrix Q as a product of elementary reflectors; If Uplo = "L", the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array Tau(), represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. [out] D() Array D(LD - 1) (LD >= N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i, i). [out] E() Array E(LE - 1) (LE >= N - 1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i, i+1) if Uplo = "U", E(i) = A(i+1, i) if Uplo = "L". [out] Tau() Array Tau(LTau - 1) (LTau >= N - 1) The scalar factors of the elementary reflectors (see Further Details). [out] Info = 0: Successful exit. = -1: The argument Uplo had an illegal value. (Uplo <> "U" nor "L") = -2: The argument N had an illegal value. (N < 0) = -3: The argument A() is invalid. = -4: The argument D() is invalid. = -5: The argument E() is invalid. = -6: The argument Tau() is invalid. Further Details If Uplo = "U", the matrix Q is represented as a product of elementary reflectors Q = H(N-1) . . . H(2) H(1). Each H(i) has the form H(i) = I - tau*v*v^T where tau is a real scalar, and v is a real vector with v(i+1〜N) = 0 and v(i) = 1. v(1〜i-1) is stored on exit in A(0〜i-2, i), and tau in Tau(i-1). If Uplo = "L", the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(N-1). Each H(i) has the form H(i) = I - tau*v*v^T where tau is a real scalar, and v is a real vector with v(1〜i) = 0 and v(i+1) = 1; v(i+2〜N) is stored on exit in A(i+1〜N-1, i-1), and tau in Tau(i-1). The contents of A() on exit are illustrated by the following examples with N = 5: if Uplo = "L": if Uplo = "U": ( d e v3 v4 v5 ) ( d ) ( d e v4 v5 ) ( e d ) ( d e v5 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector v defining H(i). Reference LAPACK Example Program See examples of Dsterf, Dsteqr and Dstebz.