00001 SUBROUTINE SGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, 00002 $ INFO ) 00003 * 00004 * -- LAPACK driver routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 INTEGER INFO, LDA, LDB, LWORK, M, N, P 00011 * .. 00012 * .. Array Arguments .. 00013 REAL A( LDA, * ), B( LDB, * ), D( * ), WORK( * ), 00014 $ X( * ), Y( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * SGGGLM solves a general Gauss-Markov linear model (GLM) problem: 00021 * 00022 * minimize || y ||_2 subject to d = A*x + B*y 00023 * x 00024 * 00025 * where A is an N-by-M matrix, B is an N-by-P matrix, and d is a 00026 * given N-vector. It is assumed that M <= N <= M+P, and 00027 * 00028 * rank(A) = M and rank( A B ) = N. 00029 * 00030 * Under these assumptions, the constrained equation is always 00031 * consistent, and there is a unique solution x and a minimal 2-norm 00032 * solution y, which is obtained using a generalized QR factorization 00033 * of the matrices (A, B) given by 00034 * 00035 * A = Q*(R), B = Q*T*Z. 00036 * (0) 00037 * 00038 * In particular, if matrix B is square nonsingular, then the problem 00039 * GLM is equivalent to the following weighted linear least squares 00040 * problem 00041 * 00042 * minimize || inv(B)*(d-A*x) ||_2 00043 * x 00044 * 00045 * where inv(B) denotes the inverse of B. 00046 * 00047 * Arguments 00048 * ========= 00049 * 00050 * N (input) INTEGER 00051 * The number of rows of the matrices A and B. N >= 0. 00052 * 00053 * M (input) INTEGER 00054 * The number of columns of the matrix A. 0 <= M <= N. 00055 * 00056 * P (input) INTEGER 00057 * The number of columns of the matrix B. P >= N-M. 00058 * 00059 * A (input/output) REAL array, dimension (LDA,M) 00060 * On entry, the N-by-M matrix A. 00061 * On exit, the upper triangular part of the array A contains 00062 * the M-by-M upper triangular matrix R. 00063 * 00064 * LDA (input) INTEGER 00065 * The leading dimension of the array A. LDA >= max(1,N). 00066 * 00067 * B (input/output) REAL array, dimension (LDB,P) 00068 * On entry, the N-by-P matrix B. 00069 * On exit, if N <= P, the upper triangle of the subarray 00070 * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; 00071 * if N > P, the elements on and above the (N-P)th subdiagonal 00072 * contain the N-by-P upper trapezoidal matrix T. 00073 * 00074 * LDB (input) INTEGER 00075 * The leading dimension of the array B. LDB >= max(1,N). 00076 * 00077 * D (input/output) REAL array, dimension (N) 00078 * On entry, D is the left hand side of the GLM equation. 00079 * On exit, D is destroyed. 00080 * 00081 * X (output) REAL array, dimension (M) 00082 * Y (output) REAL array, dimension (P) 00083 * On exit, X and Y are the solutions of the GLM problem. 00084 * 00085 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) 00086 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00087 * 00088 * LWORK (input) INTEGER 00089 * The dimension of the array WORK. LWORK >= max(1,N+M+P). 00090 * For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, 00091 * where NB is an upper bound for the optimal blocksizes for 00092 * SGEQRF, SGERQF, SORMQR and SORMRQ. 00093 * 00094 * If LWORK = -1, then a workspace query is assumed; the routine 00095 * only calculates the optimal size of the WORK array, returns 00096 * this value as the first entry of the WORK array, and no error 00097 * message related to LWORK is issued by XERBLA. 00098 * 00099 * INFO (output) INTEGER 00100 * = 0: successful exit. 00101 * < 0: if INFO = -i, the i-th argument had an illegal value. 00102 * = 1: the upper triangular factor R associated with A in the 00103 * generalized QR factorization of the pair (A, B) is 00104 * singular, so that rank(A) < M; the least squares 00105 * solution could not be computed. 00106 * = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal 00107 * factor T associated with B in the generalized QR 00108 * factorization of the pair (A, B) is singular, so that 00109 * rank( A B ) < N; the least squares solution could not 00110 * be computed. 00111 * 00112 * =================================================================== 00113 * 00114 * .. Parameters .. 00115 REAL ZERO, ONE 00116 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00117 * .. 00118 * .. Local Scalars .. 00119 LOGICAL LQUERY 00120 INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3, 00121 $ NB4, NP 00122 * .. 00123 * .. External Subroutines .. 00124 EXTERNAL SCOPY, SGEMV, SGGQRF, SORMQR, SORMRQ, STRTRS, 00125 $ XERBLA 00126 * .. 00127 * .. External Functions .. 00128 INTEGER ILAENV 00129 EXTERNAL ILAENV 00130 * .. 00131 * .. Intrinsic Functions .. 00132 INTRINSIC INT, MAX, MIN 00133 * .. 00134 * .. Executable Statements .. 00135 * 00136 * Test the input parameters 00137 * 00138 INFO = 0 00139 NP = MIN( N, P ) 00140 LQUERY = ( LWORK.EQ.-1 ) 00141 IF( N.LT.0 ) THEN 00142 INFO = -1 00143 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN 00144 INFO = -2 00145 ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN 00146 INFO = -3 00147 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00148 INFO = -5 00149 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00150 INFO = -7 00151 END IF 00152 * 00153 * Calculate workspace 00154 * 00155 IF( INFO.EQ.0) THEN 00156 IF( N.EQ.0 ) THEN 00157 LWKMIN = 1 00158 LWKOPT = 1 00159 ELSE 00160 NB1 = ILAENV( 1, 'SGEQRF', ' ', N, M, -1, -1 ) 00161 NB2 = ILAENV( 1, 'SGERQF', ' ', N, M, -1, -1 ) 00162 NB3 = ILAENV( 1, 'SORMQR', ' ', N, M, P, -1 ) 00163 NB4 = ILAENV( 1, 'SORMRQ', ' ', N, M, P, -1 ) 00164 NB = MAX( NB1, NB2, NB3, NB4 ) 00165 LWKMIN = M + N + P 00166 LWKOPT = M + NP + MAX( N, P )*NB 00167 END IF 00168 WORK( 1 ) = LWKOPT 00169 * 00170 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN 00171 INFO = -12 00172 END IF 00173 END IF 00174 * 00175 IF( INFO.NE.0 ) THEN 00176 CALL XERBLA( 'SGGGLM', -INFO ) 00177 RETURN 00178 ELSE IF( LQUERY ) THEN 00179 RETURN 00180 END IF 00181 * 00182 * Quick return if possible 00183 * 00184 IF( N.EQ.0 ) 00185 $ RETURN 00186 * 00187 * Compute the GQR factorization of matrices A and B: 00188 * 00189 * Q'*A = ( R11 ) M, Q'*B*Z' = ( T11 T12 ) M 00190 * ( 0 ) N-M ( 0 T22 ) N-M 00191 * M M+P-N N-M 00192 * 00193 * where R11 and T22 are upper triangular, and Q and Z are 00194 * orthogonal. 00195 * 00196 CALL SGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ), 00197 $ WORK( M+NP+1 ), LWORK-M-NP, INFO ) 00198 LOPT = WORK( M+NP+1 ) 00199 * 00200 * Update left-hand-side vector d = Q'*d = ( d1 ) M 00201 * ( d2 ) N-M 00202 * 00203 CALL SORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D, 00204 $ MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) 00205 LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) ) 00206 * 00207 * Solve T22*y2 = d2 for y2 00208 * 00209 IF( N.GT.M ) THEN 00210 CALL STRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1, 00211 $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO ) 00212 * 00213 IF( INFO.GT.0 ) THEN 00214 INFO = 1 00215 RETURN 00216 END IF 00217 * 00218 CALL SCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 ) 00219 END IF 00220 * 00221 * Set y1 = 0 00222 * 00223 DO 10 I = 1, M + P - N 00224 Y( I ) = ZERO 00225 10 CONTINUE 00226 * 00227 * Update d1 = d1 - T12*y2 00228 * 00229 CALL SGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB, 00230 $ Y( M+P-N+1 ), 1, ONE, D, 1 ) 00231 * 00232 * Solve triangular system: R11*x = d1 00233 * 00234 IF( M.GT.0 ) THEN 00235 CALL STRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA, 00236 $ D, M, INFO ) 00237 * 00238 IF( INFO.GT.0 ) THEN 00239 INFO = 2 00240 RETURN 00241 END IF 00242 * 00243 * Copy D to X 00244 * 00245 CALL SCOPY( M, D, 1, X, 1 ) 00246 END IF 00247 * 00248 * Backward transformation y = Z'*y 00249 * 00250 CALL SORMRQ( 'Left', 'Transpose', P, 1, NP, 00251 $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y, 00252 $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) 00253 WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) ) 00254 * 00255 RETURN 00256 * 00257 * End of SGGGLM 00258 * 00259 END
1.7.2