Annotation of rpl/lapack/lapack/dtzrqf.f, revision 1.7
1.1 bertrand 1: SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
2: *
1.5 bertrand 3: * -- LAPACK routine (version 3.2.2) --
1.1 bertrand 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.5 bertrand 6: * June 2010
1.1 bertrand 7: *
8: * .. Scalar Arguments ..
9: INTEGER INFO, LDA, M, N
10: * ..
11: * .. Array Arguments ..
12: DOUBLE PRECISION A( LDA, * ), TAU( * )
13: * ..
14: *
15: * Purpose
16: * =======
17: *
18: * This routine is deprecated and has been replaced by routine DTZRZF.
19: *
20: * DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
21: * to upper triangular form by means of orthogonal transformations.
22: *
23: * The upper trapezoidal matrix A is factored as
24: *
25: * A = ( R 0 ) * Z,
26: *
27: * where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
28: * triangular matrix.
29: *
30: * Arguments
31: * =========
32: *
33: * M (input) INTEGER
34: * The number of rows of the matrix A. M >= 0.
35: *
36: * N (input) INTEGER
37: * The number of columns of the matrix A. N >= M.
38: *
39: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
40: * On entry, the leading M-by-N upper trapezoidal part of the
41: * array A must contain the matrix to be factorized.
42: * On exit, the leading M-by-M upper triangular part of A
43: * contains the upper triangular matrix R, and elements M+1 to
44: * N of the first M rows of A, with the array TAU, represent the
45: * orthogonal matrix Z as a product of M elementary reflectors.
46: *
47: * LDA (input) INTEGER
48: * The leading dimension of the array A. LDA >= max(1,M).
49: *
50: * TAU (output) DOUBLE PRECISION array, dimension (M)
51: * The scalar factors of the elementary reflectors.
52: *
53: * INFO (output) INTEGER
54: * = 0: successful exit
55: * < 0: if INFO = -i, the i-th argument had an illegal value
56: *
57: * Further Details
58: * ===============
59: *
60: * The factorization is obtained by Householder's method. The kth
61: * transformation matrix, Z( k ), which is used to introduce zeros into
62: * the ( m - k + 1 )th row of A, is given in the form
63: *
64: * Z( k ) = ( I 0 ),
65: * ( 0 T( k ) )
66: *
67: * where
68: *
69: * T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
70: * ( 0 )
71: * ( z( k ) )
72: *
73: * tau is a scalar and z( k ) is an ( n - m ) element vector.
74: * tau and z( k ) are chosen to annihilate the elements of the kth row
75: * of X.
76: *
77: * The scalar tau is returned in the kth element of TAU and the vector
78: * u( k ) in the kth row of A, such that the elements of z( k ) are
79: * in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
80: * the upper triangular part of A.
81: *
82: * Z is given by
83: *
84: * Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
85: *
86: * =====================================================================
87: *
88: * .. Parameters ..
89: DOUBLE PRECISION ONE, ZERO
90: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
91: * ..
92: * .. Local Scalars ..
93: INTEGER I, K, M1
94: * ..
95: * .. Intrinsic Functions ..
96: INTRINSIC MAX, MIN
97: * ..
98: * .. External Subroutines ..
1.5 bertrand 99: EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
1.1 bertrand 100: * ..
101: * .. Executable Statements ..
102: *
103: * Test the input parameters.
104: *
105: INFO = 0
106: IF( M.LT.0 ) THEN
107: INFO = -1
108: ELSE IF( N.LT.M ) THEN
109: INFO = -2
110: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
111: INFO = -4
112: END IF
113: IF( INFO.NE.0 ) THEN
114: CALL XERBLA( 'DTZRQF', -INFO )
115: RETURN
116: END IF
117: *
118: * Perform the factorization.
119: *
120: IF( M.EQ.0 )
121: $ RETURN
122: IF( M.EQ.N ) THEN
123: DO 10 I = 1, N
124: TAU( I ) = ZERO
125: 10 CONTINUE
126: ELSE
127: M1 = MIN( M+1, N )
128: DO 20 K = M, 1, -1
129: *
130: * Use a Householder reflection to zero the kth row of A.
131: * First set up the reflection.
132: *
1.5 bertrand 133: CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) )
1.1 bertrand 134: *
135: IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
136: *
137: * We now perform the operation A := A*P( k ).
138: *
139: * Use the first ( k - 1 ) elements of TAU to store a( k ),
140: * where a( k ) consists of the first ( k - 1 ) elements of
141: * the kth column of A. Also let B denote the first
142: * ( k - 1 ) rows of the last ( n - m ) columns of A.
143: *
144: CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
145: *
146: * Form w = a( k ) + B*z( k ) in TAU.
147: *
148: CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
149: $ LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
150: *
151: * Now form a( k ) := a( k ) - tau*w
152: * and B := B - tau*w*z( k )'.
153: *
154: CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
155: CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
156: $ A( 1, M1 ), LDA )
157: END IF
158: 20 CONTINUE
159: END IF
160: *
161: RETURN
162: *
163: * End of DTZRQF
164: *
165: END
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