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---
:name: dsytrd
:md5sum: 0d42d32d704b2719a5b5bbbe89b1d402
:category: :subroutine
:arguments:
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- a:
:type: doublereal
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- d:
:type: doublereal
:intent: output
:dims:
- n
- e:
:type: doublereal
:intent: output
:dims:
- n-1
- tau:
:type: doublereal
:intent: output
:dims:
- n-1
- work:
:type: doublereal
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DSYTRD reduces a real symmetric matrix A to real symmetric\n\
* tridiagonal form T by an orthogonal similarity transformation:\n\
* Q**T * A * Q = T.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* = 'U': Upper triangle of A is stored;\n\
* = 'L': Lower triangle of A is stored.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A. N >= 0.\n\
*\n\
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)\n\
* On entry, the symmetric matrix A. If UPLO = 'U', the leading\n\
* N-by-N upper triangular part of A contains the upper\n\
* triangular part of the matrix A, and the strictly lower\n\
* triangular part of A is not referenced. If UPLO = 'L', the\n\
* leading N-by-N lower triangular part of A contains the lower\n\
* triangular part of the matrix A, and the strictly upper\n\
* triangular part of A is not referenced.\n\
* On exit, if UPLO = 'U', the diagonal and first superdiagonal\n\
* of A are overwritten by the corresponding elements of the\n\
* tridiagonal matrix T, and the elements above the first\n\
* superdiagonal, with the array TAU, represent the orthogonal\n\
* matrix Q as a product of elementary reflectors; if UPLO\n\
* = 'L', the diagonal and first subdiagonal of A are over-\n\
* written by the corresponding elements of the tridiagonal\n\
* matrix T, and the elements below the first subdiagonal, with\n\
* the array TAU, represent the orthogonal matrix Q as a product\n\
* of elementary reflectors. See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,N).\n\
*\n\
* D (output) DOUBLE PRECISION array, dimension (N)\n\
* The diagonal elements of the tridiagonal matrix T:\n\
* D(i) = A(i,i).\n\
*\n\
* E (output) DOUBLE PRECISION array, dimension (N-1)\n\
* The off-diagonal elements of the tridiagonal matrix T:\n\
* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.\n\
*\n\
* TAU (output) DOUBLE PRECISION array, dimension (N-1)\n\
* The scalar factors of the elementary reflectors (see Further\n\
* Details).\n\
*\n\
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))\n\
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The dimension of the array WORK. LWORK >= 1.\n\
* For optimum performance LWORK >= N*NB, where NB is the\n\
* optimal blocksize.\n\
*\n\
* If LWORK = -1, then a workspace query is assumed; the routine\n\
* only calculates the optimal size of the WORK array, returns\n\
* this value as the first entry of the WORK array, and no error\n\
* message related to LWORK is issued by XERBLA.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* If UPLO = 'U', the matrix Q is represented as a product of elementary\n\
* reflectors\n\
*\n\
* Q = H(n-1) . . . H(2) H(1).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a real scalar, and v is a real vector with\n\
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in\n\
* A(1:i-1,i+1), and tau in TAU(i).\n\
*\n\
* If UPLO = 'L', the matrix Q is represented as a product of elementary\n\
* reflectors\n\
*\n\
* Q = H(1) H(2) . . . H(n-1).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a real scalar, and v is a real vector with\n\
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),\n\
* and tau in TAU(i).\n\
*\n\
* The contents of A on exit are illustrated by the following examples\n\
* with n = 5:\n\
*\n\
* if UPLO = 'U': if UPLO = 'L':\n\
*\n\
* ( d e v2 v3 v4 ) ( d )\n\
* ( d e v3 v4 ) ( e d )\n\
* ( d e v4 ) ( v1 e d )\n\
* ( d e ) ( v1 v2 e d )\n\
* ( d ) ( v1 v2 v3 e d )\n\
*\n\
* where d and e denote diagonal and off-diagonal elements of T, and vi\n\
* denotes an element of the vector defining H(i).\n\
*\n\
* =====================================================================\n\
*\n"
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