1 SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 2 $ LDZ, IFST, ILST, WORK, LWORK, INFO ) 3* 4* -- LAPACK routine (version 3.0) -- 5* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 6* Courant Institute, Argonne National Lab, and Rice University 7* June 30, 1999 8* 9* .. Scalar Arguments .. 10 LOGICAL WANTQ, WANTZ 11 INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N 12* .. 13* .. Array Arguments .. 14 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 15 $ WORK( * ), Z( LDZ, * ) 16* .. 17* 18* Purpose 19* ======= 20* 21* DTGEXC reorders the generalized real Schur decomposition of a real 22* matrix pair (A,B) using an orthogonal equivalence transformation 23* 24* (A, B) = Q * (A, B) * Z', 25* 26* so that the diagonal block of (A, B) with row index IFST is moved 27* to row ILST. 28* 29* (A, B) must be in generalized real Schur canonical form (as returned 30* by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 31* diagonal blocks. B is upper triangular. 32* 33* Optionally, the matrices Q and Z of generalized Schur vectors are 34* updated. 35* 36* Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)' 37* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)' 38* 39* 40* Arguments 41* ========= 42* 43* WANTQ (input) LOGICAL 44* .TRUE. : update the left transformation matrix Q; 45* .FALSE.: do not update Q. 46* 47* WANTZ (input) LOGICAL 48* .TRUE. : update the right transformation matrix Z; 49* .FALSE.: do not update Z. 50* 51* N (input) INTEGER 52* The order of the matrices A and B. N >= 0. 53* 54* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 55* On entry, the matrix A in generalized real Schur canonical 56* form. 57* On exit, the updated matrix A, again in generalized 58* real Schur canonical form. 59* 60* LDA (input) INTEGER 61* The leading dimension of the array A. LDA >= max(1,N). 62* 63* B (input/output) DOUBLE PRECISION array, dimension (LDB,N) 64* On entry, the matrix B in generalized real Schur canonical 65* form (A,B). 66* On exit, the updated matrix B, again in generalized 67* real Schur canonical form (A,B). 68* 69* LDB (input) INTEGER 70* The leading dimension of the array B. LDB >= max(1,N). 71* 72* Q (input/output) DOUBLE PRECISION array, dimension (LDZ,N) 73* On entry, if WANTQ = .TRUE., the orthogonal matrix Q. 74* On exit, the updated matrix Q. 75* If WANTQ = .FALSE., Q is not referenced. 76* 77* LDQ (input) INTEGER 78* The leading dimension of the array Q. LDQ >= 1. 79* If WANTQ = .TRUE., LDQ >= N. 80* 81* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) 82* On entry, if WANTZ = .TRUE., the orthogonal matrix Z. 83* On exit, the updated matrix Z. 84* If WANTZ = .FALSE., Z is not referenced. 85* 86* LDZ (input) INTEGER 87* The leading dimension of the array Z. LDZ >= 1. 88* If WANTZ = .TRUE., LDZ >= N. 89* 90* IFST (input/output) INTEGER 91* ILST (input/output) INTEGER 92* Specify the reordering of the diagonal blocks of (A, B). 93* The block with row index IFST is moved to row ILST, by a 94* sequence of swapping between adjacent blocks. 95* On exit, if IFST pointed on entry to the second row of 96* a 2-by-2 block, it is changed to point to the first row; 97* ILST always points to the first row of the block in its 98* final position (which may differ from its input value by 99* +1 or -1). 1 <= IFST, ILST <= N. 100* 101* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) 102* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 103* 104* LWORK (input) INTEGER 105* The dimension of the array WORK. LWORK >= 4*N + 16. 106* 107* If LWORK = -1, then a workspace query is assumed; the routine 108* only calculates the optimal size of the WORK array, returns 109* this value as the first entry of the WORK array, and no error 110* message related to LWORK is issued by XERBLA. 111* 112* INFO (output) INTEGER 113* =0: successful exit. 114* <0: if INFO = -i, the i-th argument had an illegal value. 115* =1: The transformed matrix pair (A, B) would be too far 116* from generalized Schur form; the problem is ill- 117* conditioned. (A, B) may have been partially reordered, 118* and ILST points to the first row of the current 119* position of the block being moved. 120* 121* Further Details 122* =============== 123* 124* Based on contributions by 125* Bo Kagstrom and Peter Poromaa, Department of Computing Science, 126* Umea University, S-901 87 Umea, Sweden. 127* 128* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the 129* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in 130* M.S. Moonen et al (eds), Linear Algebra for Large Scale and 131* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. 132* 133* ===================================================================== 134* 135* .. Parameters .. 136 DOUBLE PRECISION ZERO 137 PARAMETER ( ZERO = 0.0D+0 ) 138* .. 139* .. Local Scalars .. 140 LOGICAL LQUERY 141 INTEGER HERE, LWMIN, NBF, NBL, NBNEXT 142* .. 143* .. External Subroutines .. 144 EXTERNAL DTGEX2, XERBLA 145* .. 146* .. Intrinsic Functions .. 147 INTRINSIC MAX 148* .. 149* .. Executable Statements .. 150* 151* Decode and test input arguments. 152* 153 INFO = 0 154 LWMIN = MAX( 1, 4*N+16 ) 155 LQUERY = ( LWORK.EQ.-1 ) 156 IF( N.LT.0 ) THEN 157 INFO = -3 158 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 159 INFO = -5 160 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 161 INFO = -7 162 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. ( LDQ.LT.MAX( 1, N ) ) ) THEN 163 INFO = -9 164 ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. ( LDZ.LT.MAX( 1, N ) ) ) THEN 165 INFO = -11 166 ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN 167 INFO = -12 168 ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN 169 INFO = -13 170 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 171 INFO = -15 172 END IF 173* 174 IF( INFO.EQ.0 ) THEN 175 WORK( 1 ) = LWMIN 176 END IF 177* 178 IF( INFO.NE.0 ) THEN 179 CALL XERBLA( 'DTGEXC', -INFO ) 180 RETURN 181 ELSE IF( LQUERY ) THEN 182 RETURN 183 END IF 184* 185* Quick return if possible 186* 187 IF( N.LE.1 ) 188 $ RETURN 189* 190* Determine the first row of the specified block and find out 191* if it is 1-by-1 or 2-by-2. 192* 193 IF( IFST.GT.1 ) THEN 194 IF( A( IFST, IFST-1 ).NE.ZERO ) 195 $ IFST = IFST - 1 196 END IF 197 NBF = 1 198 IF( IFST.LT.N ) THEN 199 IF( A( IFST+1, IFST ).NE.ZERO ) 200 $ NBF = 2 201 END IF 202* 203* Determine the first row of the final block 204* and find out if it is 1-by-1 or 2-by-2. 205* 206 IF( ILST.GT.1 ) THEN 207 IF( A( ILST, ILST-1 ).NE.ZERO ) 208 $ ILST = ILST - 1 209 END IF 210 NBL = 1 211 IF( ILST.LT.N ) THEN 212 IF( A( ILST+1, ILST ).NE.ZERO ) 213 $ NBL = 2 214 END IF 215 IF( IFST.EQ.ILST ) 216 $ RETURN 217* 218 IF( IFST.LT.ILST ) THEN 219* 220* Update ILST. 221* 222 IF( NBF.EQ.2 .AND. NBL.EQ.1 ) 223 $ ILST = ILST - 1 224 IF( NBF.EQ.1 .AND. NBL.EQ.2 ) 225 $ ILST = ILST + 1 226* 227 HERE = IFST 228* 229 10 CONTINUE 230* 231* Swap with next one below. 232* 233 IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN 234* 235* Current block either 1-by-1 or 2-by-2. 236* 237 NBNEXT = 1 238 IF( HERE+NBF+1.LE.N ) THEN 239 IF( A( HERE+NBF+1, HERE+NBF ).NE.ZERO ) 240 $ NBNEXT = 2 241 END IF 242 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 243 $ LDZ, HERE, NBF, NBNEXT, WORK, LWORK, INFO ) 244 IF( INFO.NE.0 ) THEN 245 ILST = HERE 246 RETURN 247 END IF 248 HERE = HERE + NBNEXT 249* 250* Test if 2-by-2 block breaks into two 1-by-1 blocks. 251* 252 IF( NBF.EQ.2 ) THEN 253 IF( A( HERE+1, HERE ).EQ.ZERO ) 254 $ NBF = 3 255 END IF 256* 257 ELSE 258* 259* Current block consists of two 1-by-1 blocks, each of which 260* must be swapped individually. 261* 262 NBNEXT = 1 263 IF( HERE+3.LE.N ) THEN 264 IF( A( HERE+3, HERE+2 ).NE.ZERO ) 265 $ NBNEXT = 2 266 END IF 267 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 268 $ LDZ, HERE+1, 1, NBNEXT, WORK, LWORK, INFO ) 269 IF( INFO.NE.0 ) THEN 270 ILST = HERE 271 RETURN 272 END IF 273 IF( NBNEXT.EQ.1 ) THEN 274* 275* Swap two 1-by-1 blocks. 276* 277 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 278 $ LDZ, HERE, 1, 1, WORK, LWORK, INFO ) 279 IF( INFO.NE.0 ) THEN 280 ILST = HERE 281 RETURN 282 END IF 283 HERE = HERE + 1 284* 285 ELSE 286* 287* Recompute NBNEXT in case of 2-by-2 split. 288* 289 IF( A( HERE+2, HERE+1 ).EQ.ZERO ) 290 $ NBNEXT = 1 291 IF( NBNEXT.EQ.2 ) THEN 292* 293* 2-by-2 block did not split. 294* 295 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 296 $ Z, LDZ, HERE, 1, NBNEXT, WORK, LWORK, 297 $ INFO ) 298 IF( INFO.NE.0 ) THEN 299 ILST = HERE 300 RETURN 301 END IF 302 HERE = HERE + 2 303 ELSE 304* 305* 2-by-2 block did split. 306* 307 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 308 $ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) 309 IF( INFO.NE.0 ) THEN 310 ILST = HERE 311 RETURN 312 END IF 313 HERE = HERE + 1 314 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 315 $ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) 316 IF( INFO.NE.0 ) THEN 317 ILST = HERE 318 RETURN 319 END IF 320 HERE = HERE + 1 321 END IF 322* 323 END IF 324 END IF 325 IF( HERE.LT.ILST ) 326 $ GO TO 10 327 ELSE 328 HERE = IFST 329* 330 20 CONTINUE 331* 332* Swap with next one below. 333* 334 IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN 335* 336* Current block either 1-by-1 or 2-by-2. 337* 338 NBNEXT = 1 339 IF( HERE.GE.3 ) THEN 340 IF( A( HERE-1, HERE-2 ).NE.ZERO ) 341 $ NBNEXT = 2 342 END IF 343 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 344 $ LDZ, HERE-NBNEXT, NBNEXT, NBF, WORK, LWORK, 345 $ INFO ) 346 IF( INFO.NE.0 ) THEN 347 ILST = HERE 348 RETURN 349 END IF 350 HERE = HERE - NBNEXT 351* 352* Test if 2-by-2 block breaks into two 1-by-1 blocks. 353* 354 IF( NBF.EQ.2 ) THEN 355 IF( A( HERE+1, HERE ).EQ.ZERO ) 356 $ NBF = 3 357 END IF 358* 359 ELSE 360* 361* Current block consists of two 1-by-1 blocks, each of which 362* must be swapped individually. 363* 364 NBNEXT = 1 365 IF( HERE.GE.3 ) THEN 366 IF( A( HERE-1, HERE-2 ).NE.ZERO ) 367 $ NBNEXT = 2 368 END IF 369 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 370 $ LDZ, HERE-NBNEXT, NBNEXT, 1, WORK, LWORK, 371 $ INFO ) 372 IF( INFO.NE.0 ) THEN 373 ILST = HERE 374 RETURN 375 END IF 376 IF( NBNEXT.EQ.1 ) THEN 377* 378* Swap two 1-by-1 blocks. 379* 380 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, 381 $ LDZ, HERE, NBNEXT, 1, WORK, LWORK, INFO ) 382 IF( INFO.NE.0 ) THEN 383 ILST = HERE 384 RETURN 385 END IF 386 HERE = HERE - 1 387 ELSE 388* 389* Recompute NBNEXT in case of 2-by-2 split. 390* 391 IF( A( HERE, HERE-1 ).EQ.ZERO ) 392 $ NBNEXT = 1 393 IF( NBNEXT.EQ.2 ) THEN 394* 395* 2-by-2 block did not split. 396* 397 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 398 $ Z, LDZ, HERE-1, 2, 1, WORK, LWORK, INFO ) 399 IF( INFO.NE.0 ) THEN 400 ILST = HERE 401 RETURN 402 END IF 403 HERE = HERE - 2 404 ELSE 405* 406* 2-by-2 block did split. 407* 408 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 409 $ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) 410 IF( INFO.NE.0 ) THEN 411 ILST = HERE 412 RETURN 413 END IF 414 HERE = HERE - 1 415 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, 416 $ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) 417 IF( INFO.NE.0 ) THEN 418 ILST = HERE 419 RETURN 420 END IF 421 HERE = HERE - 1 422 END IF 423 END IF 424 END IF 425 IF( HERE.GT.ILST ) 426 $ GO TO 20 427 END IF 428 ILST = HERE 429 WORK( 1 ) = LWMIN 430 RETURN 431* 432* End of DTGEXC 433* 434 END 435