.TH "gehd2" 3 "Tue Jan 28 2025 00:54:31" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
gehd2 \- gehd2: reduction to Hessenberg, level 2
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBcgehd2\fP (n, ilo, ihi, a, lda, tau, work, info)"
.br
.RI "\fBCGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&. "
.ti -1c
.RI "subroutine \fBdgehd2\fP (n, ilo, ihi, a, lda, tau, work, info)"
.br
.RI "\fBDGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&. "
.ti -1c
.RI "subroutine \fBsgehd2\fP (n, ilo, ihi, a, lda, tau, work, info)"
.br
.RI "\fBSGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&. "
.ti -1c
.RI "subroutine \fBzgehd2\fP (n, ilo, ihi, a, lda, tau, work, info)"
.br
.RI "\fBZGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&. "
.in -1c
.SH "Detailed Description"
.PP
.SH "Function Documentation"
.PP
.SS "subroutine cgehd2 (integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer info)"
.PP
\fBCGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation: Q**H * A * Q = H \&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIILO\fP
.PP
.nf
ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP
.PP
.nf
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N\&. ILO and IHI are normally
set by a previous call to CGEBAL; otherwise they should be
set to 1 and N respectively\&. See Further Details\&.
1 <= ILO <= IHI <= max(1,N)\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced\&.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details)\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX array, dimension (N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) \&. \&. \&. H(ihi-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, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i)\&.
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i)\&.
.fi
.PP
.RE
.PP
.SS "subroutine dgehd2 (integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)"
.PP
\fBDGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H \&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIILO\fP
.PP
.nf
ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP
.PP
.nf
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N\&. ILO and IHI are normally
set by a previous call to DGEBAL; otherwise they should be
set to 1 and N respectively\&. See Further Details\&.
1 <= ILO <= IHI <= max(1,N)\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced\&.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details)\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) \&. \&. \&. H(ihi-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, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i)\&.
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i)\&.
.fi
.PP
.RE
.PP
.SS "subroutine sgehd2 (integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)"
.PP
\fBSGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H \&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIILO\fP
.PP
.nf
ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP
.PP
.nf
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N\&. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively\&. See Further Details\&.
1 <= ILO <= IHI <= max(1,N)\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is REAL array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced\&.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details)\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is REAL array, dimension (N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) \&. \&. \&. H(ihi-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, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i)\&.
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i)\&.
.fi
.PP
.RE
.PP
.SS "subroutine zgehd2 (integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)"
.PP
\fBZGEHD2\fP reduces a general square matrix to upper Hessenberg form using an unblocked algorithm\&.
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation: Q**H * A * Q = H \&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrix A\&. N >= 0\&.
.fi
.PP
.br
\fIILO\fP
.PP
.nf
ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP
.PP
.nf
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N\&. ILO and IHI are normally
set by a previous call to ZGEBAL; otherwise they should be
set to 1 and N respectively\&. See Further Details\&.
1 <= ILO <= IHI <= max(1,N)\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced\&.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details)\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX*16 array, dimension (N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) \&. \&. \&. H(ihi-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, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i)\&.
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i)\&.
.fi
.PP
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.