ZGEHRD(3F) ZGEHRD(3F)
ZGEHRD - reduce a complex general matrix A to upper Hessenberg form H by
a unitary similarity transformation
SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( LWORK )
ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a
unitary similarity transformation: Q' * A * Q = H .
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
IHI (input) 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.
A (input/output) 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. LDA (input) INTEGER The leading dimension of the
array A. LDA >= max(1,N).
TAU (output) COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,N). For optimum
performance LWORK >= N*NB, where NB is the optimal blocksize.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Page 1
ZGEHRD(3F) ZGEHRD(3F)
FURTHER DETAILS
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'
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).
ZGEHRD(3F) ZGEHRD(3F)
ZGEHRD - reduce a complex general matrix A to upper Hessenberg form H by
a unitary similarity transformation
SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( LWORK )
ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a
unitary similarity transformation: Q' * A * Q = H .
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
IHI (input) 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.
A (input/output) 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. LDA (input) INTEGER The leading dimension of the
array A. LDA >= max(1,N).
TAU (output) COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,N). For optimum
performance LWORK >= N*NB, where NB is the optimal blocksize.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Page 1
ZGEHRD(3F) ZGEHRD(3F)
FURTHER DETAILS
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'
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).
PPPPaaaaggggeeee 2222 [ Back ]
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