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---
:name: dgehrd
:md5sum: 9dba8e3f6472cee11825e94623b48f41
:category: :subroutine
:arguments:
- n:
:type: integer
:intent: input
- ilo:
:type: integer
:intent: input
- ihi:
:type: integer
:intent: input
- a:
:type: doublereal
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- tau:
:type: doublereal
:intent: output
:dims:
- n-1
- work:
:type: doublereal
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DGEHRD reduces a real general matrix A to upper Hessenberg form H by\n\
* an orthogonal similarity transformation: Q' * A * Q = H .\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A. N >= 0.\n\
*\n\
* ILO (input) INTEGER\n\
* IHI (input) INTEGER\n\
* It is assumed that A is already upper triangular in rows\n\
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally\n\
* set by a previous call to DGEBAL; otherwise they should be\n\
* set to 1 and N respectively. See Further Details.\n\
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.\n\
*\n\
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)\n\
* On entry, the N-by-N general matrix to be reduced.\n\
* On exit, the upper triangle and the first subdiagonal of A\n\
* are overwritten with the upper Hessenberg matrix H, and the\n\
* elements below the first subdiagonal, with the array TAU,\n\
* represent the orthogonal matrix Q as a product of elementary\n\
* reflectors. See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,N).\n\
*\n\
* TAU (output) DOUBLE PRECISION array, dimension (N-1)\n\
* The scalar factors of the elementary reflectors (see Further\n\
* Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to\n\
* zero.\n\
*\n\
* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)\n\
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The length of the array WORK. LWORK >= max(1,N).\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\
* The matrix Q is represented as a product of (ihi-ilo) elementary\n\
* reflectors\n\
*\n\
* Q = H(ilo) H(ilo+1) . . . H(ihi-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, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on\n\
* exit in A(i+2:ihi,i), and tau in TAU(i).\n\
*\n\
* The contents of A are illustrated by the following example, with\n\
* n = 7, ilo = 2 and ihi = 6:\n\
*\n\
* on entry, on exit,\n\
*\n\
* ( a a a a a a a ) ( a a h h h h a )\n\
* ( a a a a a a ) ( a h h h h a )\n\
* ( a a a a a a ) ( h h h h h h )\n\
* ( a a a a a a ) ( v2 h h h h h )\n\
* ( a a a a a a ) ( v2 v3 h h h h )\n\
* ( a a a a a a ) ( v2 v3 v4 h h h )\n\
* ( a ) ( a )\n\
*\n\
* where a denotes an element of the original matrix A, h denotes a\n\
* modified element of the upper Hessenberg matrix H, and vi denotes an\n\
* element of the vector defining H(i).\n\
*\n\
* This file is a slight modification of LAPACK-3.0's DGEHRD\n\
* subroutine incorporating improvements proposed by Quintana-Orti and\n\
* Van de Geijn (2006). (See DLAHR2.)\n\
*\n\
* =====================================================================\n\
*\n"
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