1*> \brief \b DDRVGG 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8* Definition: 9* =========== 10* 11* SUBROUTINE DDRVGG( NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, 12* THRSHN, NOUNIT, A, LDA, B, S, T, S2, T2, Q, 13* LDQ, Z, ALPHR1, ALPHI1, BETA1, ALPHR2, ALPHI2, 14* BETA2, VL, VR, WORK, LWORK, RESULT, INFO ) 15* 16* .. Scalar Arguments .. 17* INTEGER INFO, LDA, LDQ, LWORK, NOUNIT, NSIZES, NTYPES 18* DOUBLE PRECISION THRESH, THRSHN 19* .. 20* .. Array Arguments .. 21* LOGICAL DOTYPE( * ) 22* INTEGER ISEED( 4 ), NN( * ) 23* DOUBLE PRECISION A( LDA, * ), ALPHI1( * ), ALPHI2( * ), 24* $ ALPHR1( * ), ALPHR2( * ), B( LDA, * ), 25* $ BETA1( * ), BETA2( * ), Q( LDQ, * ), 26* $ RESULT( * ), S( LDA, * ), S2( LDA, * ), 27* $ T( LDA, * ), T2( LDA, * ), VL( LDQ, * ), 28* $ VR( LDQ, * ), WORK( * ), Z( LDQ, * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> DDRVGG checks the nonsymmetric generalized eigenvalue driver 38*> routines. 39*> T T T 40*> DGEGS factors A and B as Q S Z and Q T Z , where means 41*> transpose, T is upper triangular, S is in generalized Schur form 42*> (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, 43*> the 2x2 blocks corresponding to complex conjugate pairs of 44*> generalized eigenvalues), and Q and Z are orthogonal. It also 45*> computes the generalized eigenvalues (alpha(1),beta(1)), ..., 46*> (alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j) -- 47*> thus, w(j) = alpha(j)/beta(j) is a root of the generalized 48*> eigenvalue problem 49*> 50*> det( A - w(j) B ) = 0 51*> 52*> and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent 53*> problem 54*> 55*> det( m(j) A - B ) = 0 56*> 57*> DGEGV computes the generalized eigenvalues (alpha(1),beta(1)), ..., 58*> (alpha(n),beta(n)), the matrix L whose columns contain the 59*> generalized left eigenvectors l, and the matrix R whose columns 60*> contain the generalized right eigenvectors r for the pair (A,B). 61*> 62*> When DDRVGG is called, a number of matrix "sizes" ("n's") and a 63*> number of matrix "types" are specified. For each size ("n") 64*> and each type of matrix, one matrix will be generated and used 65*> to test the nonsymmetric eigenroutines. For each matrix, 7 66*> tests will be performed and compared with the threshhold THRESH: 67*> 68*> Results from DGEGS: 69*> 70*> T 71*> (1) | A - Q S Z | / ( |A| n ulp ) 72*> 73*> T 74*> (2) | B - Q T Z | / ( |B| n ulp ) 75*> 76*> T 77*> (3) | I - QQ | / ( n ulp ) 78*> 79*> T 80*> (4) | I - ZZ | / ( n ulp ) 81*> 82*> (5) maximum over j of D(j) where: 83*> 84*> if alpha(j) is real: 85*> |alpha(j) - S(j,j)| |beta(j) - T(j,j)| 86*> D(j) = ------------------------ + ----------------------- 87*> max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|) 88*> 89*> if alpha(j) is complex: 90*> | det( s S - w T ) | 91*> D(j) = --------------------------------------------------- 92*> ulp max( s norm(S), |w| norm(T) )*norm( s S - w T ) 93*> 94*> and S and T are here the 2 x 2 diagonal blocks of S and T 95*> corresponding to the j-th eigenvalue. 96*> 97*> Results from DGEGV: 98*> 99*> (6) max over all left eigenvalue/-vector pairs (beta/alpha,l) of 100*> 101*> | l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) ) 102*> 103*> where l**H is the conjugate tranpose of l. 104*> 105*> (7) max over all right eigenvalue/-vector pairs (beta/alpha,r) of 106*> 107*> | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) ) 108*> 109*> Test Matrices 110*> ---- -------- 111*> 112*> The sizes of the test matrices are specified by an array 113*> NN(1:NSIZES); the value of each element NN(j) specifies one size. 114*> The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if 115*> DOTYPE(j) is .TRUE., then matrix type "j" will be generated. 116*> Currently, the list of possible types is: 117*> 118*> (1) ( 0, 0 ) (a pair of zero matrices) 119*> 120*> (2) ( I, 0 ) (an identity and a zero matrix) 121*> 122*> (3) ( 0, I ) (an identity and a zero matrix) 123*> 124*> (4) ( I, I ) (a pair of identity matrices) 125*> 126*> t t 127*> (5) ( J , J ) (a pair of transposed Jordan blocks) 128*> 129*> t ( I 0 ) 130*> (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) 131*> ( 0 I ) ( 0 J ) 132*> and I is a k x k identity and J a (k+1)x(k+1) 133*> Jordan block; k=(N-1)/2 134*> 135*> (7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal 136*> matrix with those diagonal entries.) 137*> (8) ( I, D ) 138*> 139*> (9) ( big*D, small*I ) where "big" is near overflow and small=1/big 140*> 141*> (10) ( small*D, big*I ) 142*> 143*> (11) ( big*I, small*D ) 144*> 145*> (12) ( small*I, big*D ) 146*> 147*> (13) ( big*D, big*I ) 148*> 149*> (14) ( small*D, small*I ) 150*> 151*> (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and 152*> D2 is diag( 0, N-3, N-4,..., 1, 0, 0 ) 153*> t t 154*> (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. 155*> 156*> (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices 157*> with random O(1) entries above the diagonal 158*> and diagonal entries diag(T1) = 159*> ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = 160*> ( 0, N-3, N-4,..., 1, 0, 0 ) 161*> 162*> (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) 163*> diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) 164*> s = machine precision. 165*> 166*> (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ) 167*> diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) 168*> 169*> N-5 170*> (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) 171*> diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) 172*> 173*> (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 ) 174*> diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) 175*> where r1,..., r(N-4) are random. 176*> 177*> (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) 178*> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) 179*> 180*> (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) 181*> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) 182*> 183*> (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) 184*> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) 185*> 186*> (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) 187*> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) 188*> 189*> (26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular 190*> matrices. 191*> \endverbatim 192* 193* Arguments: 194* ========== 195* 196*> \param[in] NSIZES 197*> \verbatim 198*> NSIZES is INTEGER 199*> The number of sizes of matrices to use. If it is zero, 200*> DDRVGG does nothing. It must be at least zero. 201*> \endverbatim 202*> 203*> \param[in] NN 204*> \verbatim 205*> NN is INTEGER array, dimension (NSIZES) 206*> An array containing the sizes to be used for the matrices. 207*> Zero values will be skipped. The values must be at least 208*> zero. 209*> \endverbatim 210*> 211*> \param[in] NTYPES 212*> \verbatim 213*> NTYPES is INTEGER 214*> The number of elements in DOTYPE. If it is zero, DDRVGG 215*> does nothing. It must be at least zero. If it is MAXTYP+1 216*> and NSIZES is 1, then an additional type, MAXTYP+1 is 217*> defined, which is to use whatever matrix is in A. This 218*> is only useful if DOTYPE(1:MAXTYP) is .FALSE. and 219*> DOTYPE(MAXTYP+1) is .TRUE. . 220*> \endverbatim 221*> 222*> \param[in] DOTYPE 223*> \verbatim 224*> DOTYPE is LOGICAL array, dimension (NTYPES) 225*> If DOTYPE(j) is .TRUE., then for each size in NN a 226*> matrix of that size and of type j will be generated. 227*> If NTYPES is smaller than the maximum number of types 228*> defined (PARAMETER MAXTYP), then types NTYPES+1 through 229*> MAXTYP will not be generated. If NTYPES is larger 230*> than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) 231*> will be ignored. 232*> \endverbatim 233*> 234*> \param[in,out] ISEED 235*> \verbatim 236*> ISEED is INTEGER array, dimension (4) 237*> On entry ISEED specifies the seed of the random number 238*> generator. The array elements should be between 0 and 4095; 239*> if not they will be reduced mod 4096. Also, ISEED(4) must 240*> be odd. The random number generator uses a linear 241*> congruential sequence limited to small integers, and so 242*> should produce machine independent random numbers. The 243*> values of ISEED are changed on exit, and can be used in the 244*> next call to DDRVGG to continue the same random number 245*> sequence. 246*> \endverbatim 247*> 248*> \param[in] THRESH 249*> \verbatim 250*> THRESH is DOUBLE PRECISION 251*> A test will count as "failed" if the "error", computed as 252*> described above, exceeds THRESH. Note that the error is 253*> scaled to be O(1), so THRESH should be a reasonably small 254*> multiple of 1, e.g., 10 or 100. In particular, it should 255*> not depend on the precision (single vs. double) or the size 256*> of the matrix. It must be at least zero. 257*> \endverbatim 258*> 259*> \param[in] THRSHN 260*> \verbatim 261*> THRSHN is DOUBLE PRECISION 262*> Threshhold for reporting eigenvector normalization error. 263*> If the normalization of any eigenvector differs from 1 by 264*> more than THRSHN*ulp, then a special error message will be 265*> printed. (This is handled separately from the other tests, 266*> since only a compiler or programming error should cause an 267*> error message, at least if THRSHN is at least 5--10.) 268*> \endverbatim 269*> 270*> \param[in] NOUNIT 271*> \verbatim 272*> NOUNIT is INTEGER 273*> The FORTRAN unit number for printing out error messages 274*> (e.g., if a routine returns IINFO not equal to 0.) 275*> \endverbatim 276*> 277*> \param[in,out] A 278*> \verbatim 279*> A is DOUBLE PRECISION array, dimension 280*> (LDA, max(NN)) 281*> Used to hold the original A matrix. Used as input only 282*> if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and 283*> DOTYPE(MAXTYP+1)=.TRUE. 284*> \endverbatim 285*> 286*> \param[in] LDA 287*> \verbatim 288*> LDA is INTEGER 289*> The leading dimension of A, B, S, T, S2, and T2. 290*> It must be at least 1 and at least max( NN ). 291*> \endverbatim 292*> 293*> \param[in,out] B 294*> \verbatim 295*> B is DOUBLE PRECISION array, dimension 296*> (LDA, max(NN)) 297*> Used to hold the original B matrix. Used as input only 298*> if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and 299*> DOTYPE(MAXTYP+1)=.TRUE. 300*> \endverbatim 301*> 302*> \param[out] S 303*> \verbatim 304*> S is DOUBLE PRECISION array, dimension (LDA, max(NN)) 305*> The Schur form matrix computed from A by DGEGS. On exit, S 306*> contains the Schur form matrix corresponding to the matrix 307*> in A. 308*> \endverbatim 309*> 310*> \param[out] T 311*> \verbatim 312*> T is DOUBLE PRECISION array, dimension (LDA, max(NN)) 313*> The upper triangular matrix computed from B by DGEGS. 314*> \endverbatim 315*> 316*> \param[out] S2 317*> \verbatim 318*> S2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) 319*> The matrix computed from A by DGEGV. This will be the 320*> Schur form of some matrix related to A, but will not, in 321*> general, be the same as S. 322*> \endverbatim 323*> 324*> \param[out] T2 325*> \verbatim 326*> T2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) 327*> The matrix computed from B by DGEGV. This will be the 328*> Schur form of some matrix related to B, but will not, in 329*> general, be the same as T. 330*> \endverbatim 331*> 332*> \param[out] Q 333*> \verbatim 334*> Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)) 335*> The (left) orthogonal matrix computed by DGEGS. 336*> \endverbatim 337*> 338*> \param[in] LDQ 339*> \verbatim 340*> LDQ is INTEGER 341*> The leading dimension of Q, Z, VL, and VR. It must 342*> be at least 1 and at least max( NN ). 343*> \endverbatim 344*> 345*> \param[out] Z 346*> \verbatim 347*> Z is DOUBLE PRECISION array of 348*> dimension( LDQ, max(NN) ) 349*> The (right) orthogonal matrix computed by DGEGS. 350*> \endverbatim 351*> 352*> \param[out] ALPHR1 353*> \verbatim 354*> ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)) 355*> \endverbatim 356*> 357*> \param[out] ALPHI1 358*> \verbatim 359*> ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)) 360*> \endverbatim 361*> 362*> \param[out] BETA1 363*> \verbatim 364*> BETA1 is DOUBLE PRECISION array, dimension (max(NN)) 365*> 366*> The generalized eigenvalues of (A,B) computed by DGEGS. 367*> ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th 368*> generalized eigenvalue of the matrices in A and B. 369*> \endverbatim 370*> 371*> \param[out] ALPHR2 372*> \verbatim 373*> ALPHR2 is DOUBLE PRECISION array, dimension (max(NN)) 374*> \endverbatim 375*> 376*> \param[out] ALPHI2 377*> \verbatim 378*> ALPHI2 is DOUBLE PRECISION array, dimension (max(NN)) 379*> \endverbatim 380*> 381*> \param[out] BETA2 382*> \verbatim 383*> BETA2 is DOUBLE PRECISION array, dimension (max(NN)) 384*> 385*> The generalized eigenvalues of (A,B) computed by DGEGV. 386*> ( ALPHR2(k)+ALPHI2(k)*i ) / BETA2(k) is the k-th 387*> generalized eigenvalue of the matrices in A and B. 388*> \endverbatim 389*> 390*> \param[out] VL 391*> \verbatim 392*> VL is DOUBLE PRECISION array, dimension (LDQ, max(NN)) 393*> The (block lower triangular) left eigenvector matrix for 394*> the matrices in A and B. (See DTGEVC for the format.) 395*> \endverbatim 396*> 397*> \param[out] VR 398*> \verbatim 399*> VR is DOUBLE PRECISION array, dimension (LDQ, max(NN)) 400*> The (block upper triangular) right eigenvector matrix for 401*> the matrices in A and B. (See DTGEVC for the format.) 402*> \endverbatim 403*> 404*> \param[out] WORK 405*> \verbatim 406*> WORK is DOUBLE PRECISION array, dimension (LWORK) 407*> \endverbatim 408*> 409*> \param[in] LWORK 410*> \verbatim 411*> LWORK is INTEGER 412*> The number of entries in WORK. This must be at least 413*> 2*N + MAX( 6*N, N*(NB+1), (k+1)*(2*k+N+1) ), where 414*> "k" is the sum of the blocksize and number-of-shifts for 415*> DHGEQZ, and NB is the greatest of the blocksizes for 416*> DGEQRF, DORMQR, and DORGQR. (The blocksizes and the 417*> number-of-shifts are retrieved through calls to ILAENV.) 418*> \endverbatim 419*> 420*> \param[out] RESULT 421*> \verbatim 422*> RESULT is DOUBLE PRECISION array, dimension (15) 423*> The values computed by the tests described above. 424*> The values are currently limited to 1/ulp, to avoid 425*> overflow. 426*> \endverbatim 427*> 428*> \param[out] INFO 429*> \verbatim 430*> INFO is INTEGER 431*> = 0: successful exit 432*> < 0: if INFO = -i, the i-th argument had an illegal value. 433*> > 0: A routine returned an error code. INFO is the 434*> absolute value of the INFO value returned. 435*> \endverbatim 436* 437* Authors: 438* ======== 439* 440*> \author Univ. of Tennessee 441*> \author Univ. of California Berkeley 442*> \author Univ. of Colorado Denver 443*> \author NAG Ltd. 444* 445*> \date November 2011 446* 447*> \ingroup double_eig 448* 449* ===================================================================== 450 SUBROUTINE DDRVGG( NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, 451 $ THRSHN, NOUNIT, A, LDA, B, S, T, S2, T2, Q, 452 $ LDQ, Z, ALPHR1, ALPHI1, BETA1, ALPHR2, ALPHI2, 453 $ BETA2, VL, VR, WORK, LWORK, RESULT, INFO ) 454* 455* -- LAPACK test routine (version 3.4.0) -- 456* -- LAPACK is a software package provided by Univ. of Tennessee, -- 457* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 458* November 2011 459* 460* .. Scalar Arguments .. 461 INTEGER INFO, LDA, LDQ, LWORK, NOUNIT, NSIZES, NTYPES 462 DOUBLE PRECISION THRESH, THRSHN 463* .. 464* .. Array Arguments .. 465 LOGICAL DOTYPE( * ) 466 INTEGER ISEED( 4 ), NN( * ) 467 DOUBLE PRECISION A( LDA, * ), ALPHI1( * ), ALPHI2( * ), 468 $ ALPHR1( * ), ALPHR2( * ), B( LDA, * ), 469 $ BETA1( * ), BETA2( * ), Q( LDQ, * ), 470 $ RESULT( * ), S( LDA, * ), S2( LDA, * ), 471 $ T( LDA, * ), T2( LDA, * ), VL( LDQ, * ), 472 $ VR( LDQ, * ), WORK( * ), Z( LDQ, * ) 473* .. 474* 475* ===================================================================== 476* 477* .. Parameters .. 478 DOUBLE PRECISION ZERO, ONE 479 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 480 INTEGER MAXTYP 481 PARAMETER ( MAXTYP = 26 ) 482* .. 483* .. Local Scalars .. 484 LOGICAL BADNN, ILABAD 485 INTEGER I1, IADD, IINFO, IN, J, JC, JR, JSIZE, JTYPE, 486 $ LWKOPT, MTYPES, N, N1, NB, NBZ, NERRS, NMATS, 487 $ NMAX, NS, NTEST, NTESTT 488 DOUBLE PRECISION SAFMAX, SAFMIN, TEMP1, TEMP2, ULP, ULPINV 489* .. 490* .. Local Arrays .. 491 INTEGER IASIGN( MAXTYP ), IBSIGN( MAXTYP ), 492 $ IOLDSD( 4 ), KADD( 6 ), KAMAGN( MAXTYP ), 493 $ KATYPE( MAXTYP ), KAZERO( MAXTYP ), 494 $ KBMAGN( MAXTYP ), KBTYPE( MAXTYP ), 495 $ KBZERO( MAXTYP ), KCLASS( MAXTYP ), 496 $ KTRIAN( MAXTYP ), KZ1( 6 ), KZ2( 6 ) 497 DOUBLE PRECISION DUMMA( 4 ), RMAGN( 0: 3 ) 498* .. 499* .. External Functions .. 500 INTEGER ILAENV 501 DOUBLE PRECISION DLAMCH, DLARND 502 EXTERNAL ILAENV, DLAMCH, DLARND 503* .. 504* .. External Subroutines .. 505 EXTERNAL ALASVM, DGEGS, DGEGV, DGET51, DGET52, DGET53, 506 $ DLABAD, DLACPY, DLARFG, DLASET, DLATM4, DORM2R, 507 $ XERBLA 508* .. 509* .. Intrinsic Functions .. 510 INTRINSIC ABS, DBLE, MAX, MIN, SIGN 511* .. 512* .. Data statements .. 513 DATA KCLASS / 15*1, 10*2, 1*3 / 514 DATA KZ1 / 0, 1, 2, 1, 3, 3 / 515 DATA KZ2 / 0, 0, 1, 2, 1, 1 / 516 DATA KADD / 0, 0, 0, 0, 3, 2 / 517 DATA KATYPE / 0, 1, 0, 1, 2, 3, 4, 1, 4, 4, 1, 1, 4, 518 $ 4, 4, 2, 4, 5, 8, 7, 9, 4*4, 0 / 519 DATA KBTYPE / 0, 0, 1, 1, 2, -3, 1, 4, 1, 1, 4, 4, 520 $ 1, 1, -4, 2, -4, 8*8, 0 / 521 DATA KAZERO / 6*1, 2, 1, 2*2, 2*1, 2*2, 3, 1, 3, 522 $ 4*5, 4*3, 1 / 523 DATA KBZERO / 6*1, 1, 2, 2*1, 2*2, 2*1, 4, 1, 4, 524 $ 4*6, 4*4, 1 / 525 DATA KAMAGN / 8*1, 2, 3, 2, 3, 2, 3, 7*1, 2, 3, 3, 526 $ 2, 1 / 527 DATA KBMAGN / 8*1, 3, 2, 3, 2, 2, 3, 7*1, 3, 2, 3, 528 $ 2, 1 / 529 DATA KTRIAN / 16*0, 10*1 / 530 DATA IASIGN / 6*0, 2, 0, 2*2, 2*0, 3*2, 0, 2, 3*0, 531 $ 5*2, 0 / 532 DATA IBSIGN / 7*0, 2, 2*0, 2*2, 2*0, 2, 0, 2, 9*0 / 533* .. 534* .. Executable Statements .. 535* 536* Check for errors 537* 538 INFO = 0 539* 540 BADNN = .FALSE. 541 NMAX = 1 542 DO 10 J = 1, NSIZES 543 NMAX = MAX( NMAX, NN( J ) ) 544 IF( NN( J ).LT.0 ) 545 $ BADNN = .TRUE. 546 10 CONTINUE 547* 548* Maximum blocksize and shift -- we assume that blocksize and number 549* of shifts are monotone increasing functions of N. 550* 551 NB = MAX( 1, ILAENV( 1, 'DGEQRF', ' ', NMAX, NMAX, -1, -1 ), 552 $ ILAENV( 1, 'DORMQR', 'LT', NMAX, NMAX, NMAX, -1 ), 553 $ ILAENV( 1, 'DORGQR', ' ', NMAX, NMAX, NMAX, -1 ) ) 554 NBZ = ILAENV( 1, 'DHGEQZ', 'SII', NMAX, 1, NMAX, 0 ) 555 NS = ILAENV( 4, 'DHGEQZ', 'SII', NMAX, 1, NMAX, 0 ) 556 I1 = NBZ + NS 557 LWKOPT = 2*NMAX + MAX( 6*NMAX, NMAX*( NB+1 ), 558 $ ( 2*I1+NMAX+1 )*( I1+1 ) ) 559* 560* Check for errors 561* 562 IF( NSIZES.LT.0 ) THEN 563 INFO = -1 564 ELSE IF( BADNN ) THEN 565 INFO = -2 566 ELSE IF( NTYPES.LT.0 ) THEN 567 INFO = -3 568 ELSE IF( THRESH.LT.ZERO ) THEN 569 INFO = -6 570 ELSE IF( LDA.LE.1 .OR. LDA.LT.NMAX ) THEN 571 INFO = -10 572 ELSE IF( LDQ.LE.1 .OR. LDQ.LT.NMAX ) THEN 573 INFO = -19 574 ELSE IF( LWKOPT.GT.LWORK ) THEN 575 INFO = -30 576 END IF 577* 578 IF( INFO.NE.0 ) THEN 579 CALL XERBLA( 'DDRVGG', -INFO ) 580 RETURN 581 END IF 582* 583* Quick return if possible 584* 585 IF( NSIZES.EQ.0 .OR. NTYPES.EQ.0 ) 586 $ RETURN 587* 588 SAFMIN = DLAMCH( 'Safe minimum' ) 589 ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) 590 SAFMIN = SAFMIN / ULP 591 SAFMAX = ONE / SAFMIN 592 CALL DLABAD( SAFMIN, SAFMAX ) 593 ULPINV = ONE / ULP 594* 595* The values RMAGN(2:3) depend on N, see below. 596* 597 RMAGN( 0 ) = ZERO 598 RMAGN( 1 ) = ONE 599* 600* Loop over sizes, types 601* 602 NTESTT = 0 603 NERRS = 0 604 NMATS = 0 605* 606 DO 170 JSIZE = 1, NSIZES 607 N = NN( JSIZE ) 608 N1 = MAX( 1, N ) 609 RMAGN( 2 ) = SAFMAX*ULP / DBLE( N1 ) 610 RMAGN( 3 ) = SAFMIN*ULPINV*N1 611* 612 IF( NSIZES.NE.1 ) THEN 613 MTYPES = MIN( MAXTYP, NTYPES ) 614 ELSE 615 MTYPES = MIN( MAXTYP+1, NTYPES ) 616 END IF 617* 618 DO 160 JTYPE = 1, MTYPES 619 IF( .NOT.DOTYPE( JTYPE ) ) 620 $ GO TO 160 621 NMATS = NMATS + 1 622 NTEST = 0 623* 624* Save ISEED in case of an error. 625* 626 DO 20 J = 1, 4 627 IOLDSD( J ) = ISEED( J ) 628 20 CONTINUE 629* 630* Initialize RESULT 631* 632 DO 30 J = 1, 15 633 RESULT( J ) = ZERO 634 30 CONTINUE 635* 636* Compute A and B 637* 638* Description of control parameters: 639* 640* KZLASS: =1 means w/o rotation, =2 means w/ rotation, 641* =3 means random. 642* KATYPE: the "type" to be passed to DLATM4 for computing A. 643* KAZERO: the pattern of zeros on the diagonal for A: 644* =1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ), 645* =4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ), 646* =6: ( 0, 1, 0, xxx, 0 ). (xxx means a string of 647* non-zero entries.) 648* KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1), 649* =2: large, =3: small. 650* IASIGN: 1 if the diagonal elements of A are to be 651* multiplied by a random magnitude 1 number, =2 if 652* randomly chosen diagonal blocks are to be rotated 653* to form 2x2 blocks. 654* KBTYPE, KBZERO, KBMAGN, IBSIGN: the same, but for B. 655* KTRIAN: =0: don't fill in the upper triangle, =1: do. 656* KZ1, KZ2, KADD: used to implement KAZERO and KBZERO. 657* RMAGN: used to implement KAMAGN and KBMAGN. 658* 659 IF( MTYPES.GT.MAXTYP ) 660 $ GO TO 110 661 IINFO = 0 662 IF( KCLASS( JTYPE ).LT.3 ) THEN 663* 664* Generate A (w/o rotation) 665* 666 IF( ABS( KATYPE( JTYPE ) ).EQ.3 ) THEN 667 IN = 2*( ( N-1 ) / 2 ) + 1 668 IF( IN.NE.N ) 669 $ CALL DLASET( 'Full', N, N, ZERO, ZERO, A, LDA ) 670 ELSE 671 IN = N 672 END IF 673 CALL DLATM4( KATYPE( JTYPE ), IN, KZ1( KAZERO( JTYPE ) ), 674 $ KZ2( KAZERO( JTYPE ) ), IASIGN( JTYPE ), 675 $ RMAGN( KAMAGN( JTYPE ) ), ULP, 676 $ RMAGN( KTRIAN( JTYPE )*KAMAGN( JTYPE ) ), 2, 677 $ ISEED, A, LDA ) 678 IADD = KADD( KAZERO( JTYPE ) ) 679 IF( IADD.GT.0 .AND. IADD.LE.N ) 680 $ A( IADD, IADD ) = ONE 681* 682* Generate B (w/o rotation) 683* 684 IF( ABS( KBTYPE( JTYPE ) ).EQ.3 ) THEN 685 IN = 2*( ( N-1 ) / 2 ) + 1 686 IF( IN.NE.N ) 687 $ CALL DLASET( 'Full', N, N, ZERO, ZERO, B, LDA ) 688 ELSE 689 IN = N 690 END IF 691 CALL DLATM4( KBTYPE( JTYPE ), IN, KZ1( KBZERO( JTYPE ) ), 692 $ KZ2( KBZERO( JTYPE ) ), IBSIGN( JTYPE ), 693 $ RMAGN( KBMAGN( JTYPE ) ), ONE, 694 $ RMAGN( KTRIAN( JTYPE )*KBMAGN( JTYPE ) ), 2, 695 $ ISEED, B, LDA ) 696 IADD = KADD( KBZERO( JTYPE ) ) 697 IF( IADD.NE.0 .AND. IADD.LE.N ) 698 $ B( IADD, IADD ) = ONE 699* 700 IF( KCLASS( JTYPE ).EQ.2 .AND. N.GT.0 ) THEN 701* 702* Include rotations 703* 704* Generate Q, Z as Householder transformations times 705* a diagonal matrix. 706* 707 DO 50 JC = 1, N - 1 708 DO 40 JR = JC, N 709 Q( JR, JC ) = DLARND( 3, ISEED ) 710 Z( JR, JC ) = DLARND( 3, ISEED ) 711 40 CONTINUE 712 CALL DLARFG( N+1-JC, Q( JC, JC ), Q( JC+1, JC ), 1, 713 $ WORK( JC ) ) 714 WORK( 2*N+JC ) = SIGN( ONE, Q( JC, JC ) ) 715 Q( JC, JC ) = ONE 716 CALL DLARFG( N+1-JC, Z( JC, JC ), Z( JC+1, JC ), 1, 717 $ WORK( N+JC ) ) 718 WORK( 3*N+JC ) = SIGN( ONE, Z( JC, JC ) ) 719 Z( JC, JC ) = ONE 720 50 CONTINUE 721 Q( N, N ) = ONE 722 WORK( N ) = ZERO 723 WORK( 3*N ) = SIGN( ONE, DLARND( 2, ISEED ) ) 724 Z( N, N ) = ONE 725 WORK( 2*N ) = ZERO 726 WORK( 4*N ) = SIGN( ONE, DLARND( 2, ISEED ) ) 727* 728* Apply the diagonal matrices 729* 730 DO 70 JC = 1, N 731 DO 60 JR = 1, N 732 A( JR, JC ) = WORK( 2*N+JR )*WORK( 3*N+JC )* 733 $ A( JR, JC ) 734 B( JR, JC ) = WORK( 2*N+JR )*WORK( 3*N+JC )* 735 $ B( JR, JC ) 736 60 CONTINUE 737 70 CONTINUE 738 CALL DORM2R( 'L', 'N', N, N, N-1, Q, LDQ, WORK, A, 739 $ LDA, WORK( 2*N+1 ), IINFO ) 740 IF( IINFO.NE.0 ) 741 $ GO TO 100 742 CALL DORM2R( 'R', 'T', N, N, N-1, Z, LDQ, WORK( N+1 ), 743 $ A, LDA, WORK( 2*N+1 ), IINFO ) 744 IF( IINFO.NE.0 ) 745 $ GO TO 100 746 CALL DORM2R( 'L', 'N', N, N, N-1, Q, LDQ, WORK, B, 747 $ LDA, WORK( 2*N+1 ), IINFO ) 748 IF( IINFO.NE.0 ) 749 $ GO TO 100 750 CALL DORM2R( 'R', 'T', N, N, N-1, Z, LDQ, WORK( N+1 ), 751 $ B, LDA, WORK( 2*N+1 ), IINFO ) 752 IF( IINFO.NE.0 ) 753 $ GO TO 100 754 END IF 755 ELSE 756* 757* Random matrices 758* 759 DO 90 JC = 1, N 760 DO 80 JR = 1, N 761 A( JR, JC ) = RMAGN( KAMAGN( JTYPE ) )* 762 $ DLARND( 2, ISEED ) 763 B( JR, JC ) = RMAGN( KBMAGN( JTYPE ) )* 764 $ DLARND( 2, ISEED ) 765 80 CONTINUE 766 90 CONTINUE 767 END IF 768* 769 100 CONTINUE 770* 771 IF( IINFO.NE.0 ) THEN 772 WRITE( NOUNIT, FMT = 9999 )'Generator', IINFO, N, JTYPE, 773 $ IOLDSD 774 INFO = ABS( IINFO ) 775 RETURN 776 END IF 777* 778 110 CONTINUE 779* 780* Call DGEGS to compute H, T, Q, Z, alpha, and beta. 781* 782 CALL DLACPY( ' ', N, N, A, LDA, S, LDA ) 783 CALL DLACPY( ' ', N, N, B, LDA, T, LDA ) 784 NTEST = 1 785 RESULT( 1 ) = ULPINV 786* 787 CALL DGEGS( 'V', 'V', N, S, LDA, T, LDA, ALPHR1, ALPHI1, 788 $ BETA1, Q, LDQ, Z, LDQ, WORK, LWORK, IINFO ) 789 IF( IINFO.NE.0 ) THEN 790 WRITE( NOUNIT, FMT = 9999 )'DGEGS', IINFO, N, JTYPE, 791 $ IOLDSD 792 INFO = ABS( IINFO ) 793 GO TO 140 794 END IF 795* 796 NTEST = 4 797* 798* Do tests 1--4 799* 800 CALL DGET51( 1, N, A, LDA, S, LDA, Q, LDQ, Z, LDQ, WORK, 801 $ RESULT( 1 ) ) 802 CALL DGET51( 1, N, B, LDA, T, LDA, Q, LDQ, Z, LDQ, WORK, 803 $ RESULT( 2 ) ) 804 CALL DGET51( 3, N, B, LDA, T, LDA, Q, LDQ, Q, LDQ, WORK, 805 $ RESULT( 3 ) ) 806 CALL DGET51( 3, N, B, LDA, T, LDA, Z, LDQ, Z, LDQ, WORK, 807 $ RESULT( 4 ) ) 808* 809* Do test 5: compare eigenvalues with diagonals. 810* Also check Schur form of A. 811* 812 TEMP1 = ZERO 813* 814 DO 120 J = 1, N 815 ILABAD = .FALSE. 816 IF( ALPHI1( J ).EQ.ZERO ) THEN 817 TEMP2 = ( ABS( ALPHR1( J )-S( J, J ) ) / 818 $ MAX( SAFMIN, ABS( ALPHR1( J ) ), ABS( S( J, 819 $ J ) ) )+ABS( BETA1( J )-T( J, J ) ) / 820 $ MAX( SAFMIN, ABS( BETA1( J ) ), ABS( T( J, 821 $ J ) ) ) ) / ULP 822 IF( J.LT.N ) THEN 823 IF( S( J+1, J ).NE.ZERO ) 824 $ ILABAD = .TRUE. 825 END IF 826 IF( J.GT.1 ) THEN 827 IF( S( J, J-1 ).NE.ZERO ) 828 $ ILABAD = .TRUE. 829 END IF 830 ELSE 831 IF( ALPHI1( J ).GT.ZERO ) THEN 832 I1 = J 833 ELSE 834 I1 = J - 1 835 END IF 836 IF( I1.LE.0 .OR. I1.GE.N ) THEN 837 ILABAD = .TRUE. 838 ELSE IF( I1.LT.N-1 ) THEN 839 IF( S( I1+2, I1+1 ).NE.ZERO ) 840 $ ILABAD = .TRUE. 841 ELSE IF( I1.GT.1 ) THEN 842 IF( S( I1, I1-1 ).NE.ZERO ) 843 $ ILABAD = .TRUE. 844 END IF 845 IF( .NOT.ILABAD ) THEN 846 CALL DGET53( S( I1, I1 ), LDA, T( I1, I1 ), LDA, 847 $ BETA1( J ), ALPHR1( J ), ALPHI1( J ), 848 $ TEMP2, IINFO ) 849 IF( IINFO.GE.3 ) THEN 850 WRITE( NOUNIT, FMT = 9997 )IINFO, J, N, JTYPE, 851 $ IOLDSD 852 INFO = ABS( IINFO ) 853 END IF 854 ELSE 855 TEMP2 = ULPINV 856 END IF 857 END IF 858 TEMP1 = MAX( TEMP1, TEMP2 ) 859 IF( ILABAD ) THEN 860 WRITE( NOUNIT, FMT = 9996 )J, N, JTYPE, IOLDSD 861 END IF 862 120 CONTINUE 863 RESULT( 5 ) = TEMP1 864* 865* Call DGEGV to compute S2, T2, VL, and VR, do tests. 866* 867* Eigenvalues and Eigenvectors 868* 869 CALL DLACPY( ' ', N, N, A, LDA, S2, LDA ) 870 CALL DLACPY( ' ', N, N, B, LDA, T2, LDA ) 871 NTEST = 6 872 RESULT( 6 ) = ULPINV 873* 874 CALL DGEGV( 'V', 'V', N, S2, LDA, T2, LDA, ALPHR2, ALPHI2, 875 $ BETA2, VL, LDQ, VR, LDQ, WORK, LWORK, IINFO ) 876 IF( IINFO.NE.0 ) THEN 877 WRITE( NOUNIT, FMT = 9999 )'DGEGV', IINFO, N, JTYPE, 878 $ IOLDSD 879 INFO = ABS( IINFO ) 880 GO TO 140 881 END IF 882* 883 NTEST = 7 884* 885* Do Tests 6 and 7 886* 887 CALL DGET52( .TRUE., N, A, LDA, B, LDA, VL, LDQ, ALPHR2, 888 $ ALPHI2, BETA2, WORK, DUMMA( 1 ) ) 889 RESULT( 6 ) = DUMMA( 1 ) 890 IF( DUMMA( 2 ).GT.THRSHN ) THEN 891 WRITE( NOUNIT, FMT = 9998 )'Left', 'DGEGV', DUMMA( 2 ), 892 $ N, JTYPE, IOLDSD 893 END IF 894* 895 CALL DGET52( .FALSE., N, A, LDA, B, LDA, VR, LDQ, ALPHR2, 896 $ ALPHI2, BETA2, WORK, DUMMA( 1 ) ) 897 RESULT( 7 ) = DUMMA( 1 ) 898 IF( DUMMA( 2 ).GT.THRESH ) THEN 899 WRITE( NOUNIT, FMT = 9998 )'Right', 'DGEGV', DUMMA( 2 ), 900 $ N, JTYPE, IOLDSD 901 END IF 902* 903* Check form of Complex eigenvalues. 904* 905 DO 130 J = 1, N 906 ILABAD = .FALSE. 907 IF( ALPHI2( J ).GT.ZERO ) THEN 908 IF( J.EQ.N ) THEN 909 ILABAD = .TRUE. 910 ELSE IF( ALPHI2( J+1 ).GE.ZERO ) THEN 911 ILABAD = .TRUE. 912 END IF 913 ELSE IF( ALPHI2( J ).LT.ZERO ) THEN 914 IF( J.EQ.1 ) THEN 915 ILABAD = .TRUE. 916 ELSE IF( ALPHI2( J-1 ).LE.ZERO ) THEN 917 ILABAD = .TRUE. 918 END IF 919 END IF 920 IF( ILABAD ) THEN 921 WRITE( NOUNIT, FMT = 9996 )J, N, JTYPE, IOLDSD 922 END IF 923 130 CONTINUE 924* 925* End of Loop -- Check for RESULT(j) > THRESH 926* 927 140 CONTINUE 928* 929 NTESTT = NTESTT + NTEST 930* 931* Print out tests which fail. 932* 933 DO 150 JR = 1, NTEST 934 IF( RESULT( JR ).GE.THRESH ) THEN 935* 936* If this is the first test to fail, 937* print a header to the data file. 938* 939 IF( NERRS.EQ.0 ) THEN 940 WRITE( NOUNIT, FMT = 9995 )'DGG' 941* 942* Matrix types 943* 944 WRITE( NOUNIT, FMT = 9994 ) 945 WRITE( NOUNIT, FMT = 9993 ) 946 WRITE( NOUNIT, FMT = 9992 )'Orthogonal' 947* 948* Tests performed 949* 950 WRITE( NOUNIT, FMT = 9991 )'orthogonal', '''', 951 $ 'transpose', ( '''', J = 1, 5 ) 952* 953 END IF 954 NERRS = NERRS + 1 955 IF( RESULT( JR ).LT.10000.0D0 ) THEN 956 WRITE( NOUNIT, FMT = 9990 )N, JTYPE, IOLDSD, JR, 957 $ RESULT( JR ) 958 ELSE 959 WRITE( NOUNIT, FMT = 9989 )N, JTYPE, IOLDSD, JR, 960 $ RESULT( JR ) 961 END IF 962 END IF 963 150 CONTINUE 964* 965 160 CONTINUE 966 170 CONTINUE 967* 968* Summary 969* 970 CALL ALASVM( 'DGG', NOUNIT, NERRS, NTESTT, 0 ) 971 RETURN 972* 973 9999 FORMAT( ' DDRVGG: ', A, ' returned INFO=', I6, '.', / 9X, 'N=', 974 $ I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), I5, ')' ) 975* 976 9998 FORMAT( ' DDRVGG: ', A, ' Eigenvectors from ', A, ' incorrectly ', 977 $ 'normalized.', / ' Bits of error=', 0P, G10.3, ',', 9X, 978 $ 'N=', I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), I5, 979 $ ')' ) 980* 981 9997 FORMAT( ' DDRVGG: DGET53 returned INFO=', I1, ' for eigenvalue ', 982 $ I6, '.', / 9X, 'N=', I6, ', JTYPE=', I6, ', ISEED=(', 983 $ 3( I5, ',' ), I5, ')' ) 984* 985 9996 FORMAT( ' DDRVGG: S not in Schur form at eigenvalue ', I6, '.', 986 $ / 9X, 'N=', I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), 987 $ I5, ')' ) 988* 989 9995 FORMAT( / 1X, A3, ' -- Real Generalized eigenvalue problem driver' 990 $ ) 991* 992 9994 FORMAT( ' Matrix types (see DDRVGG for details): ' ) 993* 994 9993 FORMAT( ' Special Matrices:', 23X, 995 $ '(J''=transposed Jordan block)', 996 $ / ' 1=(0,0) 2=(I,0) 3=(0,I) 4=(I,I) 5=(J'',J'') ', 997 $ '6=(diag(J'',I), diag(I,J''))', / ' Diagonal Matrices: ( ', 998 $ 'D=diag(0,1,2,...) )', / ' 7=(D,I) 9=(large*D, small*I', 999 $ ') 11=(large*I, small*D) 13=(large*D, large*I)', / 1000 $ ' 8=(I,D) 10=(small*D, large*I) 12=(small*I, large*D) ', 1001 $ ' 14=(small*D, small*I)', / ' 15=(D, reversed D)' ) 1002 9992 FORMAT( ' Matrices Rotated by Random ', A, ' Matrices U, V:', 1003 $ / ' 16=Transposed Jordan Blocks 19=geometric ', 1004 $ 'alpha, beta=0,1', / ' 17=arithm. alpha&beta ', 1005 $ ' 20=arithmetic alpha, beta=0,1', / ' 18=clustered ', 1006 $ 'alpha, beta=0,1 21=random alpha, beta=0,1', 1007 $ / ' Large & Small Matrices:', / ' 22=(large, small) ', 1008 $ '23=(small,large) 24=(small,small) 25=(large,large)', 1009 $ / ' 26=random O(1) matrices.' ) 1010* 1011 9991 FORMAT( / ' Tests performed: (S is Schur, T is triangular, ', 1012 $ 'Q and Z are ', A, ',', / 20X, 1013 $ 'l and r are the appropriate left and right', / 19X, 1014 $ 'eigenvectors, resp., a is alpha, b is beta, and', / 19X, A, 1015 $ ' means ', A, '.)', / ' 1 = | A - Q S Z', A, 1016 $ ' | / ( |A| n ulp ) 2 = | B - Q T Z', A, 1017 $ ' | / ( |B| n ulp )', / ' 3 = | I - QQ', A, 1018 $ ' | / ( n ulp ) 4 = | I - ZZ', A, 1019 $ ' | / ( n ulp )', / 1020 $ ' 5 = difference between (alpha,beta) and diagonals of', 1021 $ ' (S,T)', / ' 6 = max | ( b A - a B )', A, 1022 $ ' l | / const. 7 = max | ( b A - a B ) r | / const.', 1023 $ / 1X ) 1024 9990 FORMAT( ' Matrix order=', I5, ', type=', I2, ', seed=', 1025 $ 4( I4, ',' ), ' result ', I3, ' is', 0P, F8.2 ) 1026 9989 FORMAT( ' Matrix order=', I5, ', type=', I2, ', seed=', 1027 $ 4( I4, ',' ), ' result ', I3, ' is', 1P, D10.3 ) 1028* 1029* End of DDRVGG 1030* 1031 END 1032