Annotation of rpl/lapack/lapack/dtgsja.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DTGSJA
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DTGSJA + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsja.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsja.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsja.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
! 22: * LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
! 23: * Q, LDQ, WORK, NCYCLE, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER JOBQ, JOBU, JOBV
! 27: * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
! 28: * $ NCYCLE, P
! 29: * DOUBLE PRECISION TOLA, TOLB
! 30: * ..
! 31: * .. Array Arguments ..
! 32: * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
! 33: * $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
! 34: * $ V( LDV, * ), WORK( * )
! 35: * ..
! 36: *
! 37: *
! 38: *> \par Purpose:
! 39: * =============
! 40: *>
! 41: *> \verbatim
! 42: *>
! 43: *> DTGSJA computes the generalized singular value decomposition (GSVD)
! 44: *> of two real upper triangular (or trapezoidal) matrices A and B.
! 45: *>
! 46: *> On entry, it is assumed that matrices A and B have the following
! 47: *> forms, which may be obtained by the preprocessing subroutine DGGSVP
! 48: *> from a general M-by-N matrix A and P-by-N matrix B:
! 49: *>
! 50: *> N-K-L K L
! 51: *> A = K ( 0 A12 A13 ) if M-K-L >= 0;
! 52: *> L ( 0 0 A23 )
! 53: *> M-K-L ( 0 0 0 )
! 54: *>
! 55: *> N-K-L K L
! 56: *> A = K ( 0 A12 A13 ) if M-K-L < 0;
! 57: *> M-K ( 0 0 A23 )
! 58: *>
! 59: *> N-K-L K L
! 60: *> B = L ( 0 0 B13 )
! 61: *> P-L ( 0 0 0 )
! 62: *>
! 63: *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
! 64: *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
! 65: *> otherwise A23 is (M-K)-by-L upper trapezoidal.
! 66: *>
! 67: *> On exit,
! 68: *>
! 69: *> U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
! 70: *>
! 71: *> where U, V and Q are orthogonal matrices.
! 72: *> R is a nonsingular upper triangular matrix, and D1 and D2 are
! 73: *> ``diagonal'' matrices, which are of the following structures:
! 74: *>
! 75: *> If M-K-L >= 0,
! 76: *>
! 77: *> K L
! 78: *> D1 = K ( I 0 )
! 79: *> L ( 0 C )
! 80: *> M-K-L ( 0 0 )
! 81: *>
! 82: *> K L
! 83: *> D2 = L ( 0 S )
! 84: *> P-L ( 0 0 )
! 85: *>
! 86: *> N-K-L K L
! 87: *> ( 0 R ) = K ( 0 R11 R12 ) K
! 88: *> L ( 0 0 R22 ) L
! 89: *>
! 90: *> where
! 91: *>
! 92: *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
! 93: *> S = diag( BETA(K+1), ... , BETA(K+L) ),
! 94: *> C**2 + S**2 = I.
! 95: *>
! 96: *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
! 97: *>
! 98: *> If M-K-L < 0,
! 99: *>
! 100: *> K M-K K+L-M
! 101: *> D1 = K ( I 0 0 )
! 102: *> M-K ( 0 C 0 )
! 103: *>
! 104: *> K M-K K+L-M
! 105: *> D2 = M-K ( 0 S 0 )
! 106: *> K+L-M ( 0 0 I )
! 107: *> P-L ( 0 0 0 )
! 108: *>
! 109: *> N-K-L K M-K K+L-M
! 110: *> ( 0 R ) = K ( 0 R11 R12 R13 )
! 111: *> M-K ( 0 0 R22 R23 )
! 112: *> K+L-M ( 0 0 0 R33 )
! 113: *>
! 114: *> where
! 115: *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
! 116: *> S = diag( BETA(K+1), ... , BETA(M) ),
! 117: *> C**2 + S**2 = I.
! 118: *>
! 119: *> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
! 120: *> ( 0 R22 R23 )
! 121: *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
! 122: *>
! 123: *> The computation of the orthogonal transformation matrices U, V or Q
! 124: *> is optional. These matrices may either be formed explicitly, or they
! 125: *> may be postmultiplied into input matrices U1, V1, or Q1.
! 126: *> \endverbatim
! 127: *
! 128: * Arguments:
! 129: * ==========
! 130: *
! 131: *> \param[in] JOBU
! 132: *> \verbatim
! 133: *> JOBU is CHARACTER*1
! 134: *> = 'U': U must contain an orthogonal matrix U1 on entry, and
! 135: *> the product U1*U is returned;
! 136: *> = 'I': U is initialized to the unit matrix, and the
! 137: *> orthogonal matrix U is returned;
! 138: *> = 'N': U is not computed.
! 139: *> \endverbatim
! 140: *>
! 141: *> \param[in] JOBV
! 142: *> \verbatim
! 143: *> JOBV is CHARACTER*1
! 144: *> = 'V': V must contain an orthogonal matrix V1 on entry, and
! 145: *> the product V1*V is returned;
! 146: *> = 'I': V is initialized to the unit matrix, and the
! 147: *> orthogonal matrix V is returned;
! 148: *> = 'N': V is not computed.
! 149: *> \endverbatim
! 150: *>
! 151: *> \param[in] JOBQ
! 152: *> \verbatim
! 153: *> JOBQ is CHARACTER*1
! 154: *> = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
! 155: *> the product Q1*Q is returned;
! 156: *> = 'I': Q is initialized to the unit matrix, and the
! 157: *> orthogonal matrix Q is returned;
! 158: *> = 'N': Q is not computed.
! 159: *> \endverbatim
! 160: *>
! 161: *> \param[in] M
! 162: *> \verbatim
! 163: *> M is INTEGER
! 164: *> The number of rows of the matrix A. M >= 0.
! 165: *> \endverbatim
! 166: *>
! 167: *> \param[in] P
! 168: *> \verbatim
! 169: *> P is INTEGER
! 170: *> The number of rows of the matrix B. P >= 0.
! 171: *> \endverbatim
! 172: *>
! 173: *> \param[in] N
! 174: *> \verbatim
! 175: *> N is INTEGER
! 176: *> The number of columns of the matrices A and B. N >= 0.
! 177: *> \endverbatim
! 178: *>
! 179: *> \param[in] K
! 180: *> \verbatim
! 181: *> K is INTEGER
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[in] L
! 185: *> \verbatim
! 186: *> L is INTEGER
! 187: *>
! 188: *> K and L specify the subblocks in the input matrices A and B:
! 189: *> A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
! 190: *> of A and B, whose GSVD is going to be computed by DTGSJA.
! 191: *> See Further Details.
! 192: *> \endverbatim
! 193: *>
! 194: *> \param[in,out] A
! 195: *> \verbatim
! 196: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 197: *> On entry, the M-by-N matrix A.
! 198: *> On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
! 199: *> matrix R or part of R. See Purpose for details.
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] LDA
! 203: *> \verbatim
! 204: *> LDA is INTEGER
! 205: *> The leading dimension of the array A. LDA >= max(1,M).
! 206: *> \endverbatim
! 207: *>
! 208: *> \param[in,out] B
! 209: *> \verbatim
! 210: *> B is DOUBLE PRECISION array, dimension (LDB,N)
! 211: *> On entry, the P-by-N matrix B.
! 212: *> On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
! 213: *> a part of R. See Purpose for details.
! 214: *> \endverbatim
! 215: *>
! 216: *> \param[in] LDB
! 217: *> \verbatim
! 218: *> LDB is INTEGER
! 219: *> The leading dimension of the array B. LDB >= max(1,P).
! 220: *> \endverbatim
! 221: *>
! 222: *> \param[in] TOLA
! 223: *> \verbatim
! 224: *> TOLA is DOUBLE PRECISION
! 225: *> \endverbatim
! 226: *>
! 227: *> \param[in] TOLB
! 228: *> \verbatim
! 229: *> TOLB is DOUBLE PRECISION
! 230: *>
! 231: *> TOLA and TOLB are the convergence criteria for the Jacobi-
! 232: *> Kogbetliantz iteration procedure. Generally, they are the
! 233: *> same as used in the preprocessing step, say
! 234: *> TOLA = max(M,N)*norm(A)*MAZHEPS,
! 235: *> TOLB = max(P,N)*norm(B)*MAZHEPS.
! 236: *> \endverbatim
! 237: *>
! 238: *> \param[out] ALPHA
! 239: *> \verbatim
! 240: *> ALPHA is DOUBLE PRECISION array, dimension (N)
! 241: *> \endverbatim
! 242: *>
! 243: *> \param[out] BETA
! 244: *> \verbatim
! 245: *> BETA is DOUBLE PRECISION array, dimension (N)
! 246: *>
! 247: *> On exit, ALPHA and BETA contain the generalized singular
! 248: *> value pairs of A and B;
! 249: *> ALPHA(1:K) = 1,
! 250: *> BETA(1:K) = 0,
! 251: *> and if M-K-L >= 0,
! 252: *> ALPHA(K+1:K+L) = diag(C),
! 253: *> BETA(K+1:K+L) = diag(S),
! 254: *> or if M-K-L < 0,
! 255: *> ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
! 256: *> BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
! 257: *> Furthermore, if K+L < N,
! 258: *> ALPHA(K+L+1:N) = 0 and
! 259: *> BETA(K+L+1:N) = 0.
! 260: *> \endverbatim
! 261: *>
! 262: *> \param[in,out] U
! 263: *> \verbatim
! 264: *> U is DOUBLE PRECISION array, dimension (LDU,M)
! 265: *> On entry, if JOBU = 'U', U must contain a matrix U1 (usually
! 266: *> the orthogonal matrix returned by DGGSVP).
! 267: *> On exit,
! 268: *> if JOBU = 'I', U contains the orthogonal matrix U;
! 269: *> if JOBU = 'U', U contains the product U1*U.
! 270: *> If JOBU = 'N', U is not referenced.
! 271: *> \endverbatim
! 272: *>
! 273: *> \param[in] LDU
! 274: *> \verbatim
! 275: *> LDU is INTEGER
! 276: *> The leading dimension of the array U. LDU >= max(1,M) if
! 277: *> JOBU = 'U'; LDU >= 1 otherwise.
! 278: *> \endverbatim
! 279: *>
! 280: *> \param[in,out] V
! 281: *> \verbatim
! 282: *> V is DOUBLE PRECISION array, dimension (LDV,P)
! 283: *> On entry, if JOBV = 'V', V must contain a matrix V1 (usually
! 284: *> the orthogonal matrix returned by DGGSVP).
! 285: *> On exit,
! 286: *> if JOBV = 'I', V contains the orthogonal matrix V;
! 287: *> if JOBV = 'V', V contains the product V1*V.
! 288: *> If JOBV = 'N', V is not referenced.
! 289: *> \endverbatim
! 290: *>
! 291: *> \param[in] LDV
! 292: *> \verbatim
! 293: *> LDV is INTEGER
! 294: *> The leading dimension of the array V. LDV >= max(1,P) if
! 295: *> JOBV = 'V'; LDV >= 1 otherwise.
! 296: *> \endverbatim
! 297: *>
! 298: *> \param[in,out] Q
! 299: *> \verbatim
! 300: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
! 301: *> On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
! 302: *> the orthogonal matrix returned by DGGSVP).
! 303: *> On exit,
! 304: *> if JOBQ = 'I', Q contains the orthogonal matrix Q;
! 305: *> if JOBQ = 'Q', Q contains the product Q1*Q.
! 306: *> If JOBQ = 'N', Q is not referenced.
! 307: *> \endverbatim
! 308: *>
! 309: *> \param[in] LDQ
! 310: *> \verbatim
! 311: *> LDQ is INTEGER
! 312: *> The leading dimension of the array Q. LDQ >= max(1,N) if
! 313: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
! 314: *> \endverbatim
! 315: *>
! 316: *> \param[out] WORK
! 317: *> \verbatim
! 318: *> WORK is DOUBLE PRECISION array, dimension (2*N)
! 319: *> \endverbatim
! 320: *>
! 321: *> \param[out] NCYCLE
! 322: *> \verbatim
! 323: *> NCYCLE is INTEGER
! 324: *> The number of cycles required for convergence.
! 325: *> \endverbatim
! 326: *>
! 327: *> \param[out] INFO
! 328: *> \verbatim
! 329: *> INFO is INTEGER
! 330: *> = 0: successful exit
! 331: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 332: *> = 1: the procedure does not converge after MAXIT cycles.
! 333: *> \endverbatim
! 334: *>
! 335: *> \verbatim
! 336: *> Internal Parameters
! 337: *> ===================
! 338: *>
! 339: *> MAXIT INTEGER
! 340: *> MAXIT specifies the total loops that the iterative procedure
! 341: *> may take. If after MAXIT cycles, the routine fails to
! 342: *> converge, we return INFO = 1.
! 343: *> \endverbatim
! 344: *
! 345: * Authors:
! 346: * ========
! 347: *
! 348: *> \author Univ. of Tennessee
! 349: *> \author Univ. of California Berkeley
! 350: *> \author Univ. of Colorado Denver
! 351: *> \author NAG Ltd.
! 352: *
! 353: *> \date November 2011
! 354: *
! 355: *> \ingroup doubleOTHERcomputational
! 356: *
! 357: *> \par Further Details:
! 358: * =====================
! 359: *>
! 360: *> \verbatim
! 361: *>
! 362: *> DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
! 363: *> min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
! 364: *> matrix B13 to the form:
! 365: *>
! 366: *> U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
! 367: *>
! 368: *> where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
! 369: *> of Z. C1 and S1 are diagonal matrices satisfying
! 370: *>
! 371: *> C1**2 + S1**2 = I,
! 372: *>
! 373: *> and R1 is an L-by-L nonsingular upper triangular matrix.
! 374: *> \endverbatim
! 375: *>
! 376: * =====================================================================
1.1 bertrand 377: SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
378: $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
379: $ Q, LDQ, WORK, NCYCLE, INFO )
380: *
1.9 ! bertrand 381: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 382: * -- LAPACK is a software package provided by Univ. of Tennessee, --
383: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 384: * November 2011
1.1 bertrand 385: *
386: * .. Scalar Arguments ..
387: CHARACTER JOBQ, JOBU, JOBV
388: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
389: $ NCYCLE, P
390: DOUBLE PRECISION TOLA, TOLB
391: * ..
392: * .. Array Arguments ..
393: DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
394: $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
395: $ V( LDV, * ), WORK( * )
396: * ..
397: *
398: * =====================================================================
399: *
400: * .. Parameters ..
401: INTEGER MAXIT
402: PARAMETER ( MAXIT = 40 )
403: DOUBLE PRECISION ZERO, ONE
404: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
405: * ..
406: * .. Local Scalars ..
407: *
408: LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
409: INTEGER I, J, KCYCLE
410: DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
411: $ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
412: * ..
413: * .. External Functions ..
414: LOGICAL LSAME
415: EXTERNAL LSAME
416: * ..
417: * .. External Subroutines ..
418: EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
419: $ DSCAL, XERBLA
420: * ..
421: * .. Intrinsic Functions ..
422: INTRINSIC ABS, MAX, MIN
423: * ..
424: * .. Executable Statements ..
425: *
426: * Decode and test the input parameters
427: *
428: INITU = LSAME( JOBU, 'I' )
429: WANTU = INITU .OR. LSAME( JOBU, 'U' )
430: *
431: INITV = LSAME( JOBV, 'I' )
432: WANTV = INITV .OR. LSAME( JOBV, 'V' )
433: *
434: INITQ = LSAME( JOBQ, 'I' )
435: WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
436: *
437: INFO = 0
438: IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
439: INFO = -1
440: ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
441: INFO = -2
442: ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
443: INFO = -3
444: ELSE IF( M.LT.0 ) THEN
445: INFO = -4
446: ELSE IF( P.LT.0 ) THEN
447: INFO = -5
448: ELSE IF( N.LT.0 ) THEN
449: INFO = -6
450: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
451: INFO = -10
452: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
453: INFO = -12
454: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
455: INFO = -18
456: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
457: INFO = -20
458: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
459: INFO = -22
460: END IF
461: IF( INFO.NE.0 ) THEN
462: CALL XERBLA( 'DTGSJA', -INFO )
463: RETURN
464: END IF
465: *
466: * Initialize U, V and Q, if necessary
467: *
468: IF( INITU )
469: $ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
470: IF( INITV )
471: $ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
472: IF( INITQ )
473: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
474: *
475: * Loop until convergence
476: *
477: UPPER = .FALSE.
478: DO 40 KCYCLE = 1, MAXIT
479: *
480: UPPER = .NOT.UPPER
481: *
482: DO 20 I = 1, L - 1
483: DO 10 J = I + 1, L
484: *
485: A1 = ZERO
486: A2 = ZERO
487: A3 = ZERO
488: IF( K+I.LE.M )
489: $ A1 = A( K+I, N-L+I )
490: IF( K+J.LE.M )
491: $ A3 = A( K+J, N-L+J )
492: *
493: B1 = B( I, N-L+I )
494: B3 = B( J, N-L+J )
495: *
496: IF( UPPER ) THEN
497: IF( K+I.LE.M )
498: $ A2 = A( K+I, N-L+J )
499: B2 = B( I, N-L+J )
500: ELSE
501: IF( K+J.LE.M )
502: $ A2 = A( K+J, N-L+I )
503: B2 = B( J, N-L+I )
504: END IF
505: *
506: CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
507: $ CSV, SNV, CSQ, SNQ )
508: *
1.8 bertrand 509: * Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
1.1 bertrand 510: *
511: IF( K+J.LE.M )
512: $ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
513: $ LDA, CSU, SNU )
514: *
1.8 bertrand 515: * Update I-th and J-th rows of matrix B: V**T *B
1.1 bertrand 516: *
517: CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
518: $ CSV, SNV )
519: *
520: * Update (N-L+I)-th and (N-L+J)-th columns of matrices
521: * A and B: A*Q and B*Q
522: *
523: CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
524: $ A( 1, N-L+I ), 1, CSQ, SNQ )
525: *
526: CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
527: $ SNQ )
528: *
529: IF( UPPER ) THEN
530: IF( K+I.LE.M )
531: $ A( K+I, N-L+J ) = ZERO
532: B( I, N-L+J ) = ZERO
533: ELSE
534: IF( K+J.LE.M )
535: $ A( K+J, N-L+I ) = ZERO
536: B( J, N-L+I ) = ZERO
537: END IF
538: *
539: * Update orthogonal matrices U, V, Q, if desired.
540: *
541: IF( WANTU .AND. K+J.LE.M )
542: $ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
543: $ SNU )
544: *
545: IF( WANTV )
546: $ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
547: *
548: IF( WANTQ )
549: $ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
550: $ SNQ )
551: *
552: 10 CONTINUE
553: 20 CONTINUE
554: *
555: IF( .NOT.UPPER ) THEN
556: *
557: * The matrices A13 and B13 were lower triangular at the start
558: * of the cycle, and are now upper triangular.
559: *
560: * Convergence test: test the parallelism of the corresponding
561: * rows of A and B.
562: *
563: ERROR = ZERO
564: DO 30 I = 1, MIN( L, M-K )
565: CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
566: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
567: CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
568: ERROR = MAX( ERROR, SSMIN )
569: 30 CONTINUE
570: *
571: IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
572: $ GO TO 50
573: END IF
574: *
575: * End of cycle loop
576: *
577: 40 CONTINUE
578: *
579: * The algorithm has not converged after MAXIT cycles.
580: *
581: INFO = 1
582: GO TO 100
583: *
584: 50 CONTINUE
585: *
586: * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
587: * Compute the generalized singular value pairs (ALPHA, BETA), and
588: * set the triangular matrix R to array A.
589: *
590: DO 60 I = 1, K
591: ALPHA( I ) = ONE
592: BETA( I ) = ZERO
593: 60 CONTINUE
594: *
595: DO 70 I = 1, MIN( L, M-K )
596: *
597: A1 = A( K+I, N-L+I )
598: B1 = B( I, N-L+I )
599: *
600: IF( A1.NE.ZERO ) THEN
601: GAMMA = B1 / A1
602: *
603: * change sign if necessary
604: *
605: IF( GAMMA.LT.ZERO ) THEN
606: CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
607: IF( WANTV )
608: $ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
609: END IF
610: *
611: CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
612: $ RWK )
613: *
614: IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
615: CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
616: $ LDA )
617: ELSE
618: CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
619: $ LDB )
620: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
621: $ LDA )
622: END IF
623: *
624: ELSE
625: *
626: ALPHA( K+I ) = ZERO
627: BETA( K+I ) = ONE
628: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
629: $ LDA )
630: *
631: END IF
632: *
633: 70 CONTINUE
634: *
635: * Post-assignment
636: *
637: DO 80 I = M + 1, K + L
638: ALPHA( I ) = ZERO
639: BETA( I ) = ONE
640: 80 CONTINUE
641: *
642: IF( K+L.LT.N ) THEN
643: DO 90 I = K + L + 1, N
644: ALPHA( I ) = ZERO
645: BETA( I ) = ZERO
646: 90 CONTINUE
647: END IF
648: *
649: 100 CONTINUE
650: NCYCLE = KCYCLE
651: RETURN
652: *
653: * End of DTGSJA
654: *
655: END
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