1DGGGLM(1) LAPACK driver routine (version 3.2) DGGGLM(1)
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6 DGGGLM - solves 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
20 x
21 where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-
22 vector. It is assumed that M <= N <= M+P, and
23 rank(A) = M and rank( A B ) = N.
24 Under these assumptions, the constrained equation is always consistent,
25 and there is a unique solution x and a minimal 2-norm solution y, which
26 is obtained using a generalized QR factorization of the matrices (A, B)
27 given by
28 A = Q*(R), B = Q*T*Z.
29 (0)
30 In particular, if matrix B is square nonsingular, then the problem GLM
31 is equivalent to the following weighted linear least squares problem
32 minimize || inv(B)*(d-A*x) ||_2
33 x
34 where inv(B) denotes the inverse of B.
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37 N (input) INTEGER
38 The number of rows of the matrices A and B. N >= 0.
39 M (input) INTEGER
41 The number of columns of the matrix A. 0 <= M <= N.
42 P (input) INTEGER
44 The number of columns of the matrix B. P >= N-M.
45 A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
47 On entry, the N-by-M matrix A. On exit, the upper triangular
48 part of the array A contains the M-by-M upper triangular matrix
49 R.
50 LDA (input) INTEGER
52 The leading dimension of the array A. LDA >= max(1,N).
53 B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
55 On entry, the N-by-P matrix B. On exit, if N <= P, the upper
56 triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N
57 upper triangular matrix T; if N > P, the elements on and above
58 the (N-P)th subdiagonal contain the N-by-P upper trapezoidal
59 matrix T.
60 LDB (input) INTEGER
62 The leading dimension of the array B. LDB >= max(1,N).
63 D (input/output) DOUBLE PRECISION array, dimension (N)
65 On entry, D is the left hand side of the GLM equation. On
66 exit, D is destroyed.
67 X (output) DOUBLE PRECISION array, dimension (M)
69 Y (output) DOUBLE PRECISION array, dimension (P) On exit,
70 X and Y are the solutions of the GLM problem.
71 WORK (workspace/output) DOUBLE PRECISION array, dimension
73 (MAX(1,LWORK))
74 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
75 LWORK (input) INTEGER
77 The dimension of the array WORK. LWORK >= max(1,N+M+P). For
78 optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB
79 is an upper bound for the optimal blocksizes for DGEQRF,
80 SGERQF, DORMQR and SORMRQ. If LWORK = -1, then a workspace
81 query is assumed; the routine only calculates the optimal size
82 of the WORK array, returns this value as the first entry of the
83 WORK array, and no error message related to LWORK is issued by
84 XERBLA.
85 INFO (output) INTEGER
87 = 0: successful exit.
88 < 0: if INFO = -i, the i-th argument had an illegal value.
89 = 1: the upper triangular factor R associated with A in the
90 generalized QR factorization of the pair (A, B) is singular, so
91 that rank(A) < M; the least squares solution could not be com‐
92 puted. = 2: the bottom (N-M) by (N-M) part of the upper
93 trapezoidal factor T associated with B in the generalized QR
94 factorization of the pair (A, B) is singular, so that rank( A B
95 ) < N; the least squares solution could not be computed.
96 LAPACK driver routine (version 3.N2o)vember 2008 DGGGLM(1)