1 SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, 2 $ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO ) 3* 4* -- LAPACK auxiliary routine (version 3.0) -- 5* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 6* Courant Institute, Argonne National Lab, and Rice University 7* October 31, 1992 8* 9* .. Scalar Arguments .. 10 LOGICAL LTRANL, LTRANR 11 INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2 12 DOUBLE PRECISION SCALE, XNORM 13* .. 14* .. Array Arguments .. 15 DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), 16 $ X( LDX, * ) 17* .. 18* 19* Purpose 20* ======= 21* 22* DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in 23* 24* op(TL)*X + ISGN*X*op(TR) = SCALE*B, 25* 26* where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or 27* -1. op(T) = T or T', where T' denotes the transpose of T. 28* 29* Arguments 30* ========= 31* 32* LTRANL (input) LOGICAL 33* On entry, LTRANL specifies the op(TL): 34* = .FALSE., op(TL) = TL, 35* = .TRUE., op(TL) = TL'. 36* 37* LTRANR (input) LOGICAL 38* On entry, LTRANR specifies the op(TR): 39* = .FALSE., op(TR) = TR, 40* = .TRUE., op(TR) = TR'. 41* 42* ISGN (input) INTEGER 43* On entry, ISGN specifies the sign of the equation 44* as described before. ISGN may only be 1 or -1. 45* 46* N1 (input) INTEGER 47* On entry, N1 specifies the order of matrix TL. 48* N1 may only be 0, 1 or 2. 49* 50* N2 (input) INTEGER 51* On entry, N2 specifies the order of matrix TR. 52* N2 may only be 0, 1 or 2. 53* 54* TL (input) DOUBLE PRECISION array, dimension (LDTL,2) 55* On entry, TL contains an N1 by N1 matrix. 56* 57* LDTL (input) INTEGER 58* The leading dimension of the matrix TL. LDTL >= max(1,N1). 59* 60* TR (input) DOUBLE PRECISION array, dimension (LDTR,2) 61* On entry, TR contains an N2 by N2 matrix. 62* 63* LDTR (input) INTEGER 64* The leading dimension of the matrix TR. LDTR >= max(1,N2). 65* 66* B (input) DOUBLE PRECISION array, dimension (LDB,2) 67* On entry, the N1 by N2 matrix B contains the right-hand 68* side of the equation. 69* 70* LDB (input) INTEGER 71* The leading dimension of the matrix B. LDB >= max(1,N1). 72* 73* SCALE (output) DOUBLE PRECISION 74* On exit, SCALE contains the scale factor. SCALE is chosen 75* less than or equal to 1 to prevent the solution overflowing. 76* 77* X (output) DOUBLE PRECISION array, dimension (LDX,2) 78* On exit, X contains the N1 by N2 solution. 79* 80* LDX (input) INTEGER 81* The leading dimension of the matrix X. LDX >= max(1,N1). 82* 83* XNORM (output) DOUBLE PRECISION 84* On exit, XNORM is the infinity-norm of the solution. 85* 86* INFO (output) INTEGER 87* On exit, INFO is set to 88* 0: successful exit. 89* 1: TL and TR have too close eigenvalues, so TL or 90* TR is perturbed to get a nonsingular equation. 91* NOTE: In the interests of speed, this routine does not 92* check the inputs for errors. 93* 94* ===================================================================== 95* 96* .. Parameters .. 97 DOUBLE PRECISION ZERO, ONE 98 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 99 DOUBLE PRECISION TWO, HALF, EIGHT 100 PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 ) 101* .. 102* .. Local Scalars .. 103 LOGICAL BSWAP, XSWAP 104 INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K 105 DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1, 106 $ TEMP, U11, U12, U22, XMAX 107* .. 108* .. Local Arrays .. 109 LOGICAL BSWPIV( 4 ), XSWPIV( 4 ) 110 INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ), 111 $ LOCU22( 4 ) 112 DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 ) 113* .. 114* .. External Functions .. 115 INTEGER IDAMAX 116 DOUBLE PRECISION DLAMCH 117 EXTERNAL IDAMAX, DLAMCH 118* .. 119* .. External Subroutines .. 120 EXTERNAL DCOPY, DSWAP 121* .. 122* .. Intrinsic Functions .. 123 INTRINSIC ABS, MAX 124* .. 125* .. Data statements .. 126 DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / , 127 $ LOCU22 / 4, 3, 2, 1 / 128 DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. / 129 DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. / 130* .. 131* .. Executable Statements .. 132* 133* Do not check the input parameters for errors 134* 135 INFO = 0 136* 137* Quick return if possible 138* 139 IF( N1.EQ.0 .OR. N2.EQ.0 ) 140 $ RETURN 141* 142* Set constants to control overflow 143* 144 EPS = DLAMCH( 'P' ) 145 SMLNUM = DLAMCH( 'S' ) / EPS 146 SGN = ISGN 147* 148 K = N1 + N1 + N2 - 2 149 GO TO ( 10, 20, 30, 50 )K 150* 151* 1 by 1: TL11*X + SGN*X*TR11 = B11 152* 153 10 CONTINUE 154 TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 ) 155 BET = ABS( TAU1 ) 156 IF( BET.LE.SMLNUM ) THEN 157 TAU1 = SMLNUM 158 BET = SMLNUM 159 INFO = 1 160 END IF 161* 162 SCALE = ONE 163 GAM = ABS( B( 1, 1 ) ) 164 IF( SMLNUM*GAM.GT.BET ) 165 $ SCALE = ONE / GAM 166* 167 X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1 168 XNORM = ABS( X( 1, 1 ) ) 169 RETURN 170* 171* 1 by 2: 172* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12] 173* [TR21 TR22] 174* 175 20 CONTINUE 176* 177 SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ), 178 $ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ), 179 $ SMLNUM ) 180 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 181 TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) 182 IF( LTRANR ) THEN 183 TMP( 2 ) = SGN*TR( 2, 1 ) 184 TMP( 3 ) = SGN*TR( 1, 2 ) 185 ELSE 186 TMP( 2 ) = SGN*TR( 1, 2 ) 187 TMP( 3 ) = SGN*TR( 2, 1 ) 188 END IF 189 BTMP( 1 ) = B( 1, 1 ) 190 BTMP( 2 ) = B( 1, 2 ) 191 GO TO 40 192* 193* 2 by 1: 194* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11] 195* [TL21 TL22] [X21] [X21] [B21] 196* 197 30 CONTINUE 198 SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ), 199 $ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ), 200 $ SMLNUM ) 201 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 202 TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) 203 IF( LTRANL ) THEN 204 TMP( 2 ) = TL( 1, 2 ) 205 TMP( 3 ) = TL( 2, 1 ) 206 ELSE 207 TMP( 2 ) = TL( 2, 1 ) 208 TMP( 3 ) = TL( 1, 2 ) 209 END IF 210 BTMP( 1 ) = B( 1, 1 ) 211 BTMP( 2 ) = B( 2, 1 ) 212 40 CONTINUE 213* 214* Solve 2 by 2 system using complete pivoting. 215* Set pivots less than SMIN to SMIN. 216* 217 IPIV = IDAMAX( 4, TMP, 1 ) 218 U11 = TMP( IPIV ) 219 IF( ABS( U11 ).LE.SMIN ) THEN 220 INFO = 1 221 U11 = SMIN 222 END IF 223 U12 = TMP( LOCU12( IPIV ) ) 224 L21 = TMP( LOCL21( IPIV ) ) / U11 225 U22 = TMP( LOCU22( IPIV ) ) - U12*L21 226 XSWAP = XSWPIV( IPIV ) 227 BSWAP = BSWPIV( IPIV ) 228 IF( ABS( U22 ).LE.SMIN ) THEN 229 INFO = 1 230 U22 = SMIN 231 END IF 232 IF( BSWAP ) THEN 233 TEMP = BTMP( 2 ) 234 BTMP( 2 ) = BTMP( 1 ) - L21*TEMP 235 BTMP( 1 ) = TEMP 236 ELSE 237 BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 ) 238 END IF 239 SCALE = ONE 240 IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR. 241 $ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN 242 SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) ) 243 BTMP( 1 ) = BTMP( 1 )*SCALE 244 BTMP( 2 ) = BTMP( 2 )*SCALE 245 END IF 246 X2( 2 ) = BTMP( 2 ) / U22 247 X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 ) 248 IF( XSWAP ) THEN 249 TEMP = X2( 2 ) 250 X2( 2 ) = X2( 1 ) 251 X2( 1 ) = TEMP 252 END IF 253 X( 1, 1 ) = X2( 1 ) 254 IF( N1.EQ.1 ) THEN 255 X( 1, 2 ) = X2( 2 ) 256 XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) 257 ELSE 258 X( 2, 1 ) = X2( 2 ) 259 XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) ) 260 END IF 261 RETURN 262* 263* 2 by 2: 264* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12] 265* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22] 266* 267* Solve equivalent 4 by 4 system using complete pivoting. 268* Set pivots less than SMIN to SMIN. 269* 270 50 CONTINUE 271 SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ), 272 $ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ) 273 SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ), 274 $ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ) 275 SMIN = MAX( EPS*SMIN, SMLNUM ) 276 BTMP( 1 ) = ZERO 277 CALL DCOPY( 16, BTMP, 0, T16, 1 ) 278 T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 279 T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) 280 T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) 281 T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 ) 282 IF( LTRANL ) THEN 283 T16( 1, 2 ) = TL( 2, 1 ) 284 T16( 2, 1 ) = TL( 1, 2 ) 285 T16( 3, 4 ) = TL( 2, 1 ) 286 T16( 4, 3 ) = TL( 1, 2 ) 287 ELSE 288 T16( 1, 2 ) = TL( 1, 2 ) 289 T16( 2, 1 ) = TL( 2, 1 ) 290 T16( 3, 4 ) = TL( 1, 2 ) 291 T16( 4, 3 ) = TL( 2, 1 ) 292 END IF 293 IF( LTRANR ) THEN 294 T16( 1, 3 ) = SGN*TR( 1, 2 ) 295 T16( 2, 4 ) = SGN*TR( 1, 2 ) 296 T16( 3, 1 ) = SGN*TR( 2, 1 ) 297 T16( 4, 2 ) = SGN*TR( 2, 1 ) 298 ELSE 299 T16( 1, 3 ) = SGN*TR( 2, 1 ) 300 T16( 2, 4 ) = SGN*TR( 2, 1 ) 301 T16( 3, 1 ) = SGN*TR( 1, 2 ) 302 T16( 4, 2 ) = SGN*TR( 1, 2 ) 303 END IF 304 BTMP( 1 ) = B( 1, 1 ) 305 BTMP( 2 ) = B( 2, 1 ) 306 BTMP( 3 ) = B( 1, 2 ) 307 BTMP( 4 ) = B( 2, 2 ) 308* 309* Perform elimination 310* 311 DO 100 I = 1, 3 312 XMAX = ZERO 313 DO 70 IP = I, 4 314 DO 60 JP = I, 4 315 IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN 316 XMAX = ABS( T16( IP, JP ) ) 317 IPSV = IP 318 JPSV = JP 319 END IF 320 60 CONTINUE 321 70 CONTINUE 322 IF( IPSV.NE.I ) THEN 323 CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 ) 324 TEMP = BTMP( I ) 325 BTMP( I ) = BTMP( IPSV ) 326 BTMP( IPSV ) = TEMP 327 END IF 328 IF( JPSV.NE.I ) 329 $ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 ) 330 JPIV( I ) = JPSV 331 IF( ABS( T16( I, I ) ).LT.SMIN ) THEN 332 INFO = 1 333 T16( I, I ) = SMIN 334 END IF 335 DO 90 J = I + 1, 4 336 T16( J, I ) = T16( J, I ) / T16( I, I ) 337 BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I ) 338 DO 80 K = I + 1, 4 339 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K ) 340 80 CONTINUE 341 90 CONTINUE 342 100 CONTINUE 343 IF( ABS( T16( 4, 4 ) ).LT.SMIN ) 344 $ T16( 4, 4 ) = SMIN 345 SCALE = ONE 346 IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR. 347 $ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR. 348 $ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR. 349 $ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN 350 SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ), 351 $ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) ) 352 BTMP( 1 ) = BTMP( 1 )*SCALE 353 BTMP( 2 ) = BTMP( 2 )*SCALE 354 BTMP( 3 ) = BTMP( 3 )*SCALE 355 BTMP( 4 ) = BTMP( 4 )*SCALE 356 END IF 357 DO 120 I = 1, 4 358 K = 5 - I 359 TEMP = ONE / T16( K, K ) 360 TMP( K ) = BTMP( K )*TEMP 361 DO 110 J = K + 1, 4 362 TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J ) 363 110 CONTINUE 364 120 CONTINUE 365 DO 130 I = 1, 3 366 IF( JPIV( 4-I ).NE.4-I ) THEN 367 TEMP = TMP( 4-I ) 368 TMP( 4-I ) = TMP( JPIV( 4-I ) ) 369 TMP( JPIV( 4-I ) ) = TEMP 370 END IF 371 130 CONTINUE 372 X( 1, 1 ) = TMP( 1 ) 373 X( 2, 1 ) = TMP( 2 ) 374 X( 1, 2 ) = TMP( 3 ) 375 X( 2, 2 ) = TMP( 4 ) 376 XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ), 377 $ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) ) 378 RETURN 379* 380* End of DLASY2 381* 382 END 383