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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, 00002 $ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO ) 00003 * 00004 * -- LAPACK auxiliary 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 LOGICAL LTRANL, LTRANR 00011 INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2 00012 DOUBLE PRECISION SCALE, XNORM 00013 * .. 00014 * .. Array Arguments .. 00015 DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), 00016 $ X( LDX, * ) 00017 * .. 00018 * 00019 * Purpose 00020 * ======= 00021 * 00022 * DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in 00023 * 00024 * op(TL)*X + ISGN*X*op(TR) = SCALE*B, 00025 * 00026 * where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or 00027 * -1. op(T) = T or T**T, where T**T denotes the transpose of T. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * LTRANL (input) LOGICAL 00033 * On entry, LTRANL specifies the op(TL): 00034 * = .FALSE., op(TL) = TL, 00035 * = .TRUE., op(TL) = TL**T. 00036 * 00037 * LTRANR (input) LOGICAL 00038 * On entry, LTRANR specifies the op(TR): 00039 * = .FALSE., op(TR) = TR, 00040 * = .TRUE., op(TR) = TR**T. 00041 * 00042 * ISGN (input) INTEGER 00043 * On entry, ISGN specifies the sign of the equation 00044 * as described before. ISGN may only be 1 or -1. 00045 * 00046 * N1 (input) INTEGER 00047 * On entry, N1 specifies the order of matrix TL. 00048 * N1 may only be 0, 1 or 2. 00049 * 00050 * N2 (input) INTEGER 00051 * On entry, N2 specifies the order of matrix TR. 00052 * N2 may only be 0, 1 or 2. 00053 * 00054 * TL (input) DOUBLE PRECISION array, dimension (LDTL,2) 00055 * On entry, TL contains an N1 by N1 matrix. 00056 * 00057 * LDTL (input) INTEGER 00058 * The leading dimension of the matrix TL. LDTL >= max(1,N1). 00059 * 00060 * TR (input) DOUBLE PRECISION array, dimension (LDTR,2) 00061 * On entry, TR contains an N2 by N2 matrix. 00062 * 00063 * LDTR (input) INTEGER 00064 * The leading dimension of the matrix TR. LDTR >= max(1,N2). 00065 * 00066 * B (input) DOUBLE PRECISION array, dimension (LDB,2) 00067 * On entry, the N1 by N2 matrix B contains the right-hand 00068 * side of the equation. 00069 * 00070 * LDB (input) INTEGER 00071 * The leading dimension of the matrix B. LDB >= max(1,N1). 00072 * 00073 * SCALE (output) DOUBLE PRECISION 00074 * On exit, SCALE contains the scale factor. SCALE is chosen 00075 * less than or equal to 1 to prevent the solution overflowing. 00076 * 00077 * X (output) DOUBLE PRECISION array, dimension (LDX,2) 00078 * On exit, X contains the N1 by N2 solution. 00079 * 00080 * LDX (input) INTEGER 00081 * The leading dimension of the matrix X. LDX >= max(1,N1). 00082 * 00083 * XNORM (output) DOUBLE PRECISION 00084 * On exit, XNORM is the infinity-norm of the solution. 00085 * 00086 * INFO (output) INTEGER 00087 * On exit, INFO is set to 00088 * 0: successful exit. 00089 * 1: TL and TR have too close eigenvalues, so TL or 00090 * TR is perturbed to get a nonsingular equation. 00091 * NOTE: In the interests of speed, this routine does not 00092 * check the inputs for errors. 00093 * 00094 * ===================================================================== 00095 * 00096 * .. Parameters .. 00097 DOUBLE PRECISION ZERO, ONE 00098 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00099 DOUBLE PRECISION TWO, HALF, EIGHT 00100 PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 ) 00101 * .. 00102 * .. Local Scalars .. 00103 LOGICAL BSWAP, XSWAP 00104 INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K 00105 DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1, 00106 $ TEMP, U11, U12, U22, XMAX 00107 * .. 00108 * .. Local Arrays .. 00109 LOGICAL BSWPIV( 4 ), XSWPIV( 4 ) 00110 INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ), 00111 $ LOCU22( 4 ) 00112 DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 ) 00113 * .. 00114 * .. External Functions .. 00115 INTEGER IDAMAX 00116 DOUBLE PRECISION DLAMCH 00117 EXTERNAL IDAMAX, DLAMCH 00118 * .. 00119 * .. External Subroutines .. 00120 EXTERNAL DCOPY, DSWAP 00121 * .. 00122 * .. Intrinsic Functions .. 00123 INTRINSIC ABS, MAX 00124 * .. 00125 * .. Data statements .. 00126 DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / , 00127 $ LOCU22 / 4, 3, 2, 1 / 00128 DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. / 00129 DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. / 00130 * .. 00131 * .. Executable Statements .. 00132 * 00133 * Do not check the input parameters for errors 00134 * 00135 INFO = 0 00136 * 00137 * Quick return if possible 00138 * 00139 IF( N1.EQ.0 .OR. N2.EQ.0 ) 00140 $ RETURN 00141 * 00142 * Set constants to control overflow 00143 * 00144 EPS = DLAMCH( 'P' ) 00145 SMLNUM = DLAMCH( 'S' ) / EPS 00146 SGN = ISGN 00147 * 00148 K = N1 + N1 + N2 - 2 00149 GO TO ( 10, 20, 30, 50 )K 00150 * 00151 * 1 by 1: TL11*X + SGN*X*TR11 = B11 00152 * 00153 10 CONTINUE 00154 TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 ) 00155 BET = ABS( TAU1 ) 00156 IF( BET.LE.SMLNUM ) THEN 00157 TAU1 = SMLNUM 00158 BET = SMLNUM 00159 INFO = 1 00160 END IF 00161 * 00162 SCALE = ONE 00163 GAM = ABS( B( 1, 1 ) ) 00164 IF( SMLNUM*GAM.GT.BET ) 00165 $ SCALE = ONE / GAM 00166 * 00167 X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1 00168 XNORM = ABS( X( 1, 1 ) ) 00169 RETURN 00170 * 00171 * 1 by 2: 00172 * TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12] 00173 * [TR21 TR22] 00174 * 00175 20 CONTINUE 00176 * 00177 SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ), 00178 $ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ), 00179 $ SMLNUM ) 00180 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 00181 TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) 00182 IF( LTRANR ) THEN 00183 TMP( 2 ) = SGN*TR( 2, 1 ) 00184 TMP( 3 ) = SGN*TR( 1, 2 ) 00185 ELSE 00186 TMP( 2 ) = SGN*TR( 1, 2 ) 00187 TMP( 3 ) = SGN*TR( 2, 1 ) 00188 END IF 00189 BTMP( 1 ) = B( 1, 1 ) 00190 BTMP( 2 ) = B( 1, 2 ) 00191 GO TO 40 00192 * 00193 * 2 by 1: 00194 * op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11] 00195 * [TL21 TL22] [X21] [X21] [B21] 00196 * 00197 30 CONTINUE 00198 SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ), 00199 $ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ), 00200 $ SMLNUM ) 00201 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 00202 TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) 00203 IF( LTRANL ) THEN 00204 TMP( 2 ) = TL( 1, 2 ) 00205 TMP( 3 ) = TL( 2, 1 ) 00206 ELSE 00207 TMP( 2 ) = TL( 2, 1 ) 00208 TMP( 3 ) = TL( 1, 2 ) 00209 END IF 00210 BTMP( 1 ) = B( 1, 1 ) 00211 BTMP( 2 ) = B( 2, 1 ) 00212 40 CONTINUE 00213 * 00214 * Solve 2 by 2 system using complete pivoting. 00215 * Set pivots less than SMIN to SMIN. 00216 * 00217 IPIV = IDAMAX( 4, TMP, 1 ) 00218 U11 = TMP( IPIV ) 00219 IF( ABS( U11 ).LE.SMIN ) THEN 00220 INFO = 1 00221 U11 = SMIN 00222 END IF 00223 U12 = TMP( LOCU12( IPIV ) ) 00224 L21 = TMP( LOCL21( IPIV ) ) / U11 00225 U22 = TMP( LOCU22( IPIV ) ) - U12*L21 00226 XSWAP = XSWPIV( IPIV ) 00227 BSWAP = BSWPIV( IPIV ) 00228 IF( ABS( U22 ).LE.SMIN ) THEN 00229 INFO = 1 00230 U22 = SMIN 00231 END IF 00232 IF( BSWAP ) THEN 00233 TEMP = BTMP( 2 ) 00234 BTMP( 2 ) = BTMP( 1 ) - L21*TEMP 00235 BTMP( 1 ) = TEMP 00236 ELSE 00237 BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 ) 00238 END IF 00239 SCALE = ONE 00240 IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR. 00241 $ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN 00242 SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) ) 00243 BTMP( 1 ) = BTMP( 1 )*SCALE 00244 BTMP( 2 ) = BTMP( 2 )*SCALE 00245 END IF 00246 X2( 2 ) = BTMP( 2 ) / U22 00247 X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 ) 00248 IF( XSWAP ) THEN 00249 TEMP = X2( 2 ) 00250 X2( 2 ) = X2( 1 ) 00251 X2( 1 ) = TEMP 00252 END IF 00253 X( 1, 1 ) = X2( 1 ) 00254 IF( N1.EQ.1 ) THEN 00255 X( 1, 2 ) = X2( 2 ) 00256 XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) 00257 ELSE 00258 X( 2, 1 ) = X2( 2 ) 00259 XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) ) 00260 END IF 00261 RETURN 00262 * 00263 * 2 by 2: 00264 * op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12] 00265 * [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22] 00266 * 00267 * Solve equivalent 4 by 4 system using complete pivoting. 00268 * Set pivots less than SMIN to SMIN. 00269 * 00270 50 CONTINUE 00271 SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ), 00272 $ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ) 00273 SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ), 00274 $ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ) 00275 SMIN = MAX( EPS*SMIN, SMLNUM ) 00276 BTMP( 1 ) = ZERO 00277 CALL DCOPY( 16, BTMP, 0, T16, 1 ) 00278 T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) 00279 T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) 00280 T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) 00281 T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 ) 00282 IF( LTRANL ) THEN 00283 T16( 1, 2 ) = TL( 2, 1 ) 00284 T16( 2, 1 ) = TL( 1, 2 ) 00285 T16( 3, 4 ) = TL( 2, 1 ) 00286 T16( 4, 3 ) = TL( 1, 2 ) 00287 ELSE 00288 T16( 1, 2 ) = TL( 1, 2 ) 00289 T16( 2, 1 ) = TL( 2, 1 ) 00290 T16( 3, 4 ) = TL( 1, 2 ) 00291 T16( 4, 3 ) = TL( 2, 1 ) 00292 END IF 00293 IF( LTRANR ) THEN 00294 T16( 1, 3 ) = SGN*TR( 1, 2 ) 00295 T16( 2, 4 ) = SGN*TR( 1, 2 ) 00296 T16( 3, 1 ) = SGN*TR( 2, 1 ) 00297 T16( 4, 2 ) = SGN*TR( 2, 1 ) 00298 ELSE 00299 T16( 1, 3 ) = SGN*TR( 2, 1 ) 00300 T16( 2, 4 ) = SGN*TR( 2, 1 ) 00301 T16( 3, 1 ) = SGN*TR( 1, 2 ) 00302 T16( 4, 2 ) = SGN*TR( 1, 2 ) 00303 END IF 00304 BTMP( 1 ) = B( 1, 1 ) 00305 BTMP( 2 ) = B( 2, 1 ) 00306 BTMP( 3 ) = B( 1, 2 ) 00307 BTMP( 4 ) = B( 2, 2 ) 00308 * 00309 * Perform elimination 00310 * 00311 DO 100 I = 1, 3 00312 XMAX = ZERO 00313 DO 70 IP = I, 4 00314 DO 60 JP = I, 4 00315 IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN 00316 XMAX = ABS( T16( IP, JP ) ) 00317 IPSV = IP 00318 JPSV = JP 00319 END IF 00320 60 CONTINUE 00321 70 CONTINUE 00322 IF( IPSV.NE.I ) THEN 00323 CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 ) 00324 TEMP = BTMP( I ) 00325 BTMP( I ) = BTMP( IPSV ) 00326 BTMP( IPSV ) = TEMP 00327 END IF 00328 IF( JPSV.NE.I ) 00329 $ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 ) 00330 JPIV( I ) = JPSV 00331 IF( ABS( T16( I, I ) ).LT.SMIN ) THEN 00332 INFO = 1 00333 T16( I, I ) = SMIN 00334 END IF 00335 DO 90 J = I + 1, 4 00336 T16( J, I ) = T16( J, I ) / T16( I, I ) 00337 BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I ) 00338 DO 80 K = I + 1, 4 00339 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K ) 00340 80 CONTINUE 00341 90 CONTINUE 00342 100 CONTINUE 00343 IF( ABS( T16( 4, 4 ) ).LT.SMIN ) 00344 $ T16( 4, 4 ) = SMIN 00345 SCALE = ONE 00346 IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR. 00347 $ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR. 00348 $ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR. 00349 $ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN 00350 SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ), 00351 $ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) ) 00352 BTMP( 1 ) = BTMP( 1 )*SCALE 00353 BTMP( 2 ) = BTMP( 2 )*SCALE 00354 BTMP( 3 ) = BTMP( 3 )*SCALE 00355 BTMP( 4 ) = BTMP( 4 )*SCALE 00356 END IF 00357 DO 120 I = 1, 4 00358 K = 5 - I 00359 TEMP = ONE / T16( K, K ) 00360 TMP( K ) = BTMP( K )*TEMP 00361 DO 110 J = K + 1, 4 00362 TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J ) 00363 110 CONTINUE 00364 120 CONTINUE 00365 DO 130 I = 1, 3 00366 IF( JPIV( 4-I ).NE.4-I ) THEN 00367 TEMP = TMP( 4-I ) 00368 TMP( 4-I ) = TMP( JPIV( 4-I ) ) 00369 TMP( JPIV( 4-I ) ) = TEMP 00370 END IF 00371 130 CONTINUE 00372 X( 1, 1 ) = TMP( 1 ) 00373 X( 2, 1 ) = TMP( 2 ) 00374 X( 1, 2 ) = TMP( 3 ) 00375 X( 2, 2 ) = TMP( 4 ) 00376 XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ), 00377 $ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) ) 00378 RETURN 00379 * 00380 * End of DLASY2 00381 * 00382 END
1.7.3 
