1DGGGLM(1) LAPACK driver routine (version 3.1) DGGGLM(1)
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6 DGGGLM - a general Gauss-Markov linear model (GLM) problem
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9 SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO
10 )
11 INTEGER INFO, LDA, LDB, LWORK, M, N, P
13 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( *
15 ), X( * ), Y( * )
16
18 DGGGLM solves a general Gauss-Markov linear model (GLM) problem:
19 minimize || y ||_2 subject to d = A*x + B*y
21 x
22 where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-
24 vector. It is assumed that M <= N <= M+P, and
25 rank(A) = M and rank( A B ) = N.
27 Under these assumptions, the constrained equation is always consistent,
29 and there is a unique solution x and a minimal 2-norm solution y, which
30 is obtained using a generalized QR factorization of the matrices (A, B)
31 given by
32 A = Q*(R), B = Q*T*Z.
34 (0)
35 In particular, if matrix B is square nonsingular, then the problem GLM
37 is equivalent to the following weighted linear least squares problem
38 minimize || inv(B)*(d-A*x) ||_2
40 x
41 where inv(B) denotes the inverse of B.
43
46 N (input) INTEGER
47 The number of rows of the matrices A and B. N >= 0.
48 M (input) INTEGER
50 The number of columns of the matrix A. 0 <= M <= N.
51 P (input) INTEGER
53 The number of columns of the matrix B. P >= N-M.
54 A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
56 On entry, the N-by-M matrix A. On exit, the upper triangular
57 part of the array A contains the M-by-M upper triangular matrix
58 R.
59 LDA (input) INTEGER
61 The leading dimension of the array A. LDA >= max(1,N).
62 B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
64 On entry, the N-by-P matrix B. On exit, if N <= P, the upper
65 triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N
66 upper triangular matrix T; if N > P, the elements on and above
67 the (N-P)th subdiagonal contain the N-by-P upper trapezoidal
68 matrix T.
69 LDB (input) INTEGER
71 The leading dimension of the array B. LDB >= max(1,N).
72 D (input/output) DOUBLE PRECISION array, dimension (N)
74 On entry, D is the left hand side of the GLM equation. On
75 exit, D is destroyed.
76 X (output) DOUBLE PRECISION array, dimension (M)
78 Y (output) DOUBLE PRECISION array, dimension (P) On exit,
79 X and Y are the solutions of the GLM problem.
80 WORK (workspace/output) DOUBLE PRECISION array, dimension
82 (MAX(1,LWORK))
83 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
84 LWORK (input) INTEGER
86 The dimension of the array WORK. LWORK >= max(1,N+M+P). For
87 optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB
88 is an upper bound for the optimal blocksizes for DGEQRF,
89 SGERQF, DORMQR and SORMRQ.
90 If LWORK = -1, then a workspace query is assumed; the routine
92 only calculates the optimal size of the WORK array, returns
93 this value as the first entry of the WORK array, and no error
94 message related to LWORK is issued by XERBLA.
95 INFO (output) INTEGER
97 = 0: successful exit.
98 < 0: if INFO = -i, the i-th argument had an illegal value.
99 = 1: the upper triangular factor R associated with A in the
100 generalized QR factorization of the pair (A, B) is singular, so
101 that rank(A) < M; the least squares solution could not be com‐
102 puted. = 2: the bottom (N-M) by (N-M) part of the upper
103 trapezoidal factor T associated with B in the generalized QR
104 factorization of the pair (A, B) is singular, so that rank( A B
105 ) < N; the least squares solution could not be computed.
106 LAPACK driver routine (version 3.N1o)vember 2006 DGGGLM(1)