1 SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 INTEGER L, LDA, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
13 * ..
14 *
15 * Purpose
16 * =======
17 *
18 * DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
19 * [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
20 * of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
21 * matrix and, R and A1 are M-by-M upper triangular matrices.
22 *
23 * Arguments
24 * =========
25 *
26 * M (input) INTEGER
27 * The number of rows of the matrix A. M >= 0.
28 *
29 * N (input) INTEGER
30 * The number of columns of the matrix A. N >= 0.
31 *
32 * L (input) INTEGER
33 * The number of columns of the matrix A containing the
34 * meaningful part of the Householder vectors. N-M >= L >= 0.
35 *
36 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
37 * On entry, the leading M-by-N upper trapezoidal part of the
38 * array A must contain the matrix to be factorized.
39 * On exit, the leading M-by-M upper triangular part of A
40 * contains the upper triangular matrix R, and elements N-L+1 to
41 * N of the first M rows of A, with the array TAU, represent the
42 * orthogonal matrix Z as a product of M elementary reflectors.
43 *
44 * LDA (input) INTEGER
45 * The leading dimension of the array A. LDA >= max(1,M).
46 *
47 * TAU (output) DOUBLE PRECISION array, dimension (M)
48 * The scalar factors of the elementary reflectors.
49 *
50 * WORK (workspace) DOUBLE PRECISION array, dimension (M)
51 *
52 * Further Details
53 * ===============
54 *
55 * Based on contributions by
56 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
57 *
58 * The factorization is obtained by Householder's method. The kth
59 * transformation matrix, Z( k ), which is used to introduce zeros into
60 * the ( m - k + 1 )th row of A, is given in the form
61 *
62 * Z( k ) = ( I 0 ),
63 * ( 0 T( k ) )
64 *
65 * where
66 *
67 * T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
68 * ( 0 )
69 * ( z( k ) )
70 *
71 * tau is a scalar and z( k ) is an l element vector. tau and z( k )
72 * are chosen to annihilate the elements of the kth row of A2.
73 *
74 * The scalar tau is returned in the kth element of TAU and the vector
75 * u( k ) in the kth row of A2, such that the elements of z( k ) are
76 * in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
77 * the upper triangular part of A1.
78 *
79 * Z is given by
80 *
81 * Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
82 *
83 * =====================================================================
84 *
85 * .. Parameters ..
86 DOUBLE PRECISION ZERO
87 PARAMETER ( ZERO = 0.0D+0 )
88 * ..
89 * .. Local Scalars ..
90 INTEGER I
91 * ..
92 * .. External Subroutines ..
93 EXTERNAL DLARFG, DLARZ
94 * ..
95 * .. Executable Statements ..
96 *
97 * Test the input arguments
98 *
99 * Quick return if possible
100 *
101 IF( M.EQ.0 ) THEN
102 RETURN
103 ELSE IF( M.EQ.N ) THEN
104 DO 10 I = 1, N
105 TAU( I ) = ZERO
106 10 CONTINUE
107 RETURN
108 END IF
109 *
110 DO 20 I = M, 1, -1
111 *
112 * Generate elementary reflector H(i) to annihilate
113 * [ A(i,i) A(i,n-l+1:n) ]
114 *
115 CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
116 *
117 * Apply H(i) to A(1:i-1,i:n) from the right
118 *
119 CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
120 $ TAU( I ), A( 1, I ), LDA, WORK )
121 *
122 20 CONTINUE
123 *
124 RETURN
125 *
126 * End of DLATRZ
127 *
128 END
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 INTEGER L, LDA, M, N
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
13 * ..
14 *
15 * Purpose
16 * =======
17 *
18 * DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
19 * [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
20 * of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
21 * matrix and, R and A1 are M-by-M upper triangular matrices.
22 *
23 * Arguments
24 * =========
25 *
26 * M (input) INTEGER
27 * The number of rows of the matrix A. M >= 0.
28 *
29 * N (input) INTEGER
30 * The number of columns of the matrix A. N >= 0.
31 *
32 * L (input) INTEGER
33 * The number of columns of the matrix A containing the
34 * meaningful part of the Householder vectors. N-M >= L >= 0.
35 *
36 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
37 * On entry, the leading M-by-N upper trapezoidal part of the
38 * array A must contain the matrix to be factorized.
39 * On exit, the leading M-by-M upper triangular part of A
40 * contains the upper triangular matrix R, and elements N-L+1 to
41 * N of the first M rows of A, with the array TAU, represent the
42 * orthogonal matrix Z as a product of M elementary reflectors.
43 *
44 * LDA (input) INTEGER
45 * The leading dimension of the array A. LDA >= max(1,M).
46 *
47 * TAU (output) DOUBLE PRECISION array, dimension (M)
48 * The scalar factors of the elementary reflectors.
49 *
50 * WORK (workspace) DOUBLE PRECISION array, dimension (M)
51 *
52 * Further Details
53 * ===============
54 *
55 * Based on contributions by
56 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
57 *
58 * The factorization is obtained by Householder's method. The kth
59 * transformation matrix, Z( k ), which is used to introduce zeros into
60 * the ( m - k + 1 )th row of A, is given in the form
61 *
62 * Z( k ) = ( I 0 ),
63 * ( 0 T( k ) )
64 *
65 * where
66 *
67 * T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
68 * ( 0 )
69 * ( z( k ) )
70 *
71 * tau is a scalar and z( k ) is an l element vector. tau and z( k )
72 * are chosen to annihilate the elements of the kth row of A2.
73 *
74 * The scalar tau is returned in the kth element of TAU and the vector
75 * u( k ) in the kth row of A2, such that the elements of z( k ) are
76 * in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
77 * the upper triangular part of A1.
78 *
79 * Z is given by
80 *
81 * Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
82 *
83 * =====================================================================
84 *
85 * .. Parameters ..
86 DOUBLE PRECISION ZERO
87 PARAMETER ( ZERO = 0.0D+0 )
88 * ..
89 * .. Local Scalars ..
90 INTEGER I
91 * ..
92 * .. External Subroutines ..
93 EXTERNAL DLARFG, DLARZ
94 * ..
95 * .. Executable Statements ..
96 *
97 * Test the input arguments
98 *
99 * Quick return if possible
100 *
101 IF( M.EQ.0 ) THEN
102 RETURN
103 ELSE IF( M.EQ.N ) THEN
104 DO 10 I = 1, N
105 TAU( I ) = ZERO
106 10 CONTINUE
107 RETURN
108 END IF
109 *
110 DO 20 I = M, 1, -1
111 *
112 * Generate elementary reflector H(i) to annihilate
113 * [ A(i,i) A(i,n-l+1:n) ]
114 *
115 CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
116 *
117 * Apply H(i) to A(1:i-1,i:n) from the right
118 *
119 CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
120 $ TAU( I ), A( 1, I ), LDA, WORK )
121 *
122 20 CONTINUE
123 *
124 RETURN
125 *
126 * End of DLATRZ
127 *
128 END