zhbtrd zhetrd zhptrd zpteqr zstedc zstein zstemr zsteqr zungtr zunmtr zupgtr zupmtr

zhetrd() void zhetrd ( char  uplo, int  n, int  lda, doublecomplex  a[], double  d[], double  e[], doublecomplex  tau[], doublecomplex  work[], int  lwork, int *  info  ) Reduces a complex Hermitian matrix to tridiagonal form Purpose This routine reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q^H * 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] lda Leading dimension of the two dimensional array a[][]. (lda >= max(1, n)) [in,out] a[][] Array a[la][lda] (la >= n) [in] The Hermitian 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 unitary 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 unitary matrix Q as a product of elementary reflectors. See Further Details. [out] d[] Array d[ld] (ld >= n) The diagonal elements of the tridiagonal matrix T: d[i] = a[i][i]. [out] e[] Array e[le] (le >= n - 1) The off-diagonal elements of the tridiagonal matrix T: e[i] = a[i+1][i] if uplo = 'U', e[i] = a[i][i+1] if uplo = 'L'. [out] tau[] Array tau[ltau] (ltau >= n - 1) The scalar factors of the elementary reflectors (see Further Details). [out] work[] Array work[lwork] Work array. On exit, if info = 0, work[0] returns the optimal lwork. [in] lwork The size of work[]. (lwork >= 1) For optimum performance lwork >= n*nb, where nb is the optimal blocksize. If lwork = -1, then a workspace query is assumed. The routine only calculates the optimal size of the work[] array, and returns the value in work[0]. [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 lda had an illegal value. (lda < max(1, n)) = -9: The argument lwork had an illegal value. (lwork too small) 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^H where tau is a complex scalar, and v is a complex vector with v(i+1〜n) = 0 and v(i) = 1. v(1〜i-1) is stored on exit in a[i][0〜i-2], 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^H where tau is a complex scalar, and v is a complex vector with v(1〜i) = 0 and v(i+1) = 1; v(i+2〜n) is stored on exit in a[i-1][i+1〜n-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