Annotation of rpl/lapack/lapack/dtgex2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
! 2: $ LDZ, J1, N1, N2, WORK, LWORK, INFO )
! 3: *
! 4: * -- LAPACK auxiliary routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: LOGICAL WANTQ, WANTZ
! 11: INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
! 12: * ..
! 13: * .. Array Arguments ..
! 14: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 15: $ WORK( * ), Z( LDZ, * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
! 22: * of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
! 23: * (A, B) by an orthogonal equivalence transformation.
! 24: *
! 25: * (A, B) must be in generalized real Schur canonical form (as returned
! 26: * by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
! 27: * diagonal blocks. B is upper triangular.
! 28: *
! 29: * Optionally, the matrices Q and Z of generalized Schur vectors are
! 30: * updated.
! 31: *
! 32: * Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
! 33: * Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
! 34: *
! 35: *
! 36: * Arguments
! 37: * =========
! 38: *
! 39: * WANTQ (input) LOGICAL
! 40: * .TRUE. : update the left transformation matrix Q;
! 41: * .FALSE.: do not update Q.
! 42: *
! 43: * WANTZ (input) LOGICAL
! 44: * .TRUE. : update the right transformation matrix Z;
! 45: * .FALSE.: do not update Z.
! 46: *
! 47: * N (input) INTEGER
! 48: * The order of the matrices A and B. N >= 0.
! 49: *
! 50: * A (input/output) DOUBLE PRECISION arrays, dimensions (LDA,N)
! 51: * On entry, the matrix A in the pair (A, B).
! 52: * On exit, the updated matrix A.
! 53: *
! 54: * LDA (input) INTEGER
! 55: * The leading dimension of the array A. LDA >= max(1,N).
! 56: *
! 57: * B (input/output) DOUBLE PRECISION arrays, dimensions (LDB,N)
! 58: * On entry, the matrix B in the pair (A, B).
! 59: * On exit, the updated matrix B.
! 60: *
! 61: * LDB (input) INTEGER
! 62: * The leading dimension of the array B. LDB >= max(1,N).
! 63: *
! 64: * Q (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
! 65: * On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
! 66: * On exit, the updated matrix Q.
! 67: * Not referenced if WANTQ = .FALSE..
! 68: *
! 69: * LDQ (input) INTEGER
! 70: * The leading dimension of the array Q. LDQ >= 1.
! 71: * If WANTQ = .TRUE., LDQ >= N.
! 72: *
! 73: * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
! 74: * On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
! 75: * On exit, the updated matrix Z.
! 76: * Not referenced if WANTZ = .FALSE..
! 77: *
! 78: * LDZ (input) INTEGER
! 79: * The leading dimension of the array Z. LDZ >= 1.
! 80: * If WANTZ = .TRUE., LDZ >= N.
! 81: *
! 82: * J1 (input) INTEGER
! 83: * The index to the first block (A11, B11). 1 <= J1 <= N.
! 84: *
! 85: * N1 (input) INTEGER
! 86: * The order of the first block (A11, B11). N1 = 0, 1 or 2.
! 87: *
! 88: * N2 (input) INTEGER
! 89: * The order of the second block (A22, B22). N2 = 0, 1 or 2.
! 90: *
! 91: * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
! 92: *
! 93: * LWORK (input) INTEGER
! 94: * The dimension of the array WORK.
! 95: * LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
! 96: *
! 97: * INFO (output) INTEGER
! 98: * =0: Successful exit
! 99: * >0: If INFO = 1, the transformed matrix (A, B) would be
! 100: * too far from generalized Schur form; the blocks are
! 101: * not swapped and (A, B) and (Q, Z) are unchanged.
! 102: * The problem of swapping is too ill-conditioned.
! 103: * <0: If INFO = -16: LWORK is too small. Appropriate value
! 104: * for LWORK is returned in WORK(1).
! 105: *
! 106: * Further Details
! 107: * ===============
! 108: *
! 109: * Based on contributions by
! 110: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 111: * Umea University, S-901 87 Umea, Sweden.
! 112: *
! 113: * In the current code both weak and strong stability tests are
! 114: * performed. The user can omit the strong stability test by changing
! 115: * the internal logical parameter WANDS to .FALSE.. See ref. [2] for
! 116: * details.
! 117: *
! 118: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
! 119: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
! 120: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
! 121: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
! 122: *
! 123: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
! 124: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
! 125: * Estimation: Theory, Algorithms and Software,
! 126: * Report UMINF - 94.04, Department of Computing Science, Umea
! 127: * University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
! 128: * Note 87. To appear in Numerical Algorithms, 1996.
! 129: *
! 130: * =====================================================================
! 131: * Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
! 132: * loops. Sven Hammarling, 1/5/02.
! 133: *
! 134: * .. Parameters ..
! 135: DOUBLE PRECISION ZERO, ONE
! 136: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 137: DOUBLE PRECISION TEN
! 138: PARAMETER ( TEN = 1.0D+01 )
! 139: INTEGER LDST
! 140: PARAMETER ( LDST = 4 )
! 141: LOGICAL WANDS
! 142: PARAMETER ( WANDS = .TRUE. )
! 143: * ..
! 144: * .. Local Scalars ..
! 145: LOGICAL DTRONG, WEAK
! 146: INTEGER I, IDUM, LINFO, M
! 147: DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,
! 148: $ F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS
! 149: * ..
! 150: * .. Local Arrays ..
! 151: INTEGER IWORK( LDST )
! 152: DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
! 153: $ IRCOP( LDST, LDST ), LI( LDST, LDST ),
! 154: $ LICOP( LDST, LDST ), S( LDST, LDST ),
! 155: $ SCPY( LDST, LDST ), T( LDST, LDST ),
! 156: $ TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
! 157: * ..
! 158: * .. External Functions ..
! 159: DOUBLE PRECISION DLAMCH
! 160: EXTERNAL DLAMCH
! 161: * ..
! 162: * .. External Subroutines ..
! 163: EXTERNAL DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2, DLARTG,
! 164: $ DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
! 165: $ DROT, DSCAL, DTGSY2
! 166: * ..
! 167: * .. Intrinsic Functions ..
! 168: INTRINSIC ABS, MAX, SQRT
! 169: * ..
! 170: * .. Executable Statements ..
! 171: *
! 172: INFO = 0
! 173: *
! 174: * Quick return if possible
! 175: *
! 176: IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
! 177: $ RETURN
! 178: IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
! 179: $ RETURN
! 180: M = N1 + N2
! 181: IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
! 182: INFO = -16
! 183: WORK( 1 ) = MAX( 1, N*M, M*M*2 )
! 184: RETURN
! 185: END IF
! 186: *
! 187: WEAK = .FALSE.
! 188: DTRONG = .FALSE.
! 189: *
! 190: * Make a local copy of selected block
! 191: *
! 192: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
! 193: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
! 194: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
! 195: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
! 196: *
! 197: * Compute threshold for testing acceptance of swapping.
! 198: *
! 199: EPS = DLAMCH( 'P' )
! 200: SMLNUM = DLAMCH( 'S' ) / EPS
! 201: DSCALE = ZERO
! 202: DSUM = ONE
! 203: CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
! 204: CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
! 205: CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
! 206: CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
! 207: DNORM = DSCALE*SQRT( DSUM )
! 208: THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
! 209: *
! 210: IF( M.EQ.2 ) THEN
! 211: *
! 212: * CASE 1: Swap 1-by-1 and 1-by-1 blocks.
! 213: *
! 214: * Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
! 215: * using Givens rotations and perform the swap tentatively.
! 216: *
! 217: F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
! 218: G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
! 219: SB = ABS( T( 2, 2 ) )
! 220: SA = ABS( S( 2, 2 ) )
! 221: CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
! 222: IR( 2, 1 ) = -IR( 1, 2 )
! 223: IR( 2, 2 ) = IR( 1, 1 )
! 224: CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
! 225: $ IR( 2, 1 ) )
! 226: CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
! 227: $ IR( 2, 1 ) )
! 228: IF( SA.GE.SB ) THEN
! 229: CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
! 230: $ DDUM )
! 231: ELSE
! 232: CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
! 233: $ DDUM )
! 234: END IF
! 235: CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
! 236: $ LI( 2, 1 ) )
! 237: CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
! 238: $ LI( 2, 1 ) )
! 239: LI( 2, 2 ) = LI( 1, 1 )
! 240: LI( 1, 2 ) = -LI( 2, 1 )
! 241: *
! 242: * Weak stability test:
! 243: * |S21| + |T21| <= O(EPS * F-norm((S, T)))
! 244: *
! 245: WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
! 246: WEAK = WS.LE.THRESH
! 247: IF( .NOT.WEAK )
! 248: $ GO TO 70
! 249: *
! 250: IF( WANDS ) THEN
! 251: *
! 252: * Strong stability test:
! 253: * F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B)))
! 254: *
! 255: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
! 256: $ M )
! 257: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
! 258: $ WORK, M )
! 259: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
! 260: $ WORK( M*M+1 ), M )
! 261: DSCALE = ZERO
! 262: DSUM = ONE
! 263: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
! 264: *
! 265: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
! 266: $ M )
! 267: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
! 268: $ WORK, M )
! 269: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
! 270: $ WORK( M*M+1 ), M )
! 271: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
! 272: SS = DSCALE*SQRT( DSUM )
! 273: DTRONG = SS.LE.THRESH
! 274: IF( .NOT.DTRONG )
! 275: $ GO TO 70
! 276: END IF
! 277: *
! 278: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
! 279: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
! 280: *
! 281: CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
! 282: $ IR( 2, 1 ) )
! 283: CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
! 284: $ IR( 2, 1 ) )
! 285: CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
! 286: $ LI( 1, 1 ), LI( 2, 1 ) )
! 287: CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
! 288: $ LI( 1, 1 ), LI( 2, 1 ) )
! 289: *
! 290: * Set N1-by-N2 (2,1) - blocks to ZERO.
! 291: *
! 292: A( J1+1, J1 ) = ZERO
! 293: B( J1+1, J1 ) = ZERO
! 294: *
! 295: * Accumulate transformations into Q and Z if requested.
! 296: *
! 297: IF( WANTZ )
! 298: $ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
! 299: $ IR( 2, 1 ) )
! 300: IF( WANTQ )
! 301: $ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
! 302: $ LI( 2, 1 ) )
! 303: *
! 304: * Exit with INFO = 0 if swap was successfully performed.
! 305: *
! 306: RETURN
! 307: *
! 308: ELSE
! 309: *
! 310: * CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
! 311: * and 2-by-2 blocks.
! 312: *
! 313: * Solve the generalized Sylvester equation
! 314: * S11 * R - L * S22 = SCALE * S12
! 315: * T11 * R - L * T22 = SCALE * T12
! 316: * for R and L. Solutions in LI and IR.
! 317: *
! 318: CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
! 319: CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
! 320: $ IR( N2+1, N1+1 ), LDST )
! 321: CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
! 322: $ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
! 323: $ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
! 324: $ LINFO )
! 325: *
! 326: * Compute orthogonal matrix QL:
! 327: *
! 328: * QL' * LI = [ TL ]
! 329: * [ 0 ]
! 330: * where
! 331: * LI = [ -L ]
! 332: * [ SCALE * identity(N2) ]
! 333: *
! 334: DO 10 I = 1, N2
! 335: CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
! 336: LI( N1+I, I ) = SCALE
! 337: 10 CONTINUE
! 338: CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
! 339: IF( LINFO.NE.0 )
! 340: $ GO TO 70
! 341: CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
! 342: IF( LINFO.NE.0 )
! 343: $ GO TO 70
! 344: *
! 345: * Compute orthogonal matrix RQ:
! 346: *
! 347: * IR * RQ' = [ 0 TR],
! 348: *
! 349: * where IR = [ SCALE * identity(N1), R ]
! 350: *
! 351: DO 20 I = 1, N1
! 352: IR( N2+I, I ) = SCALE
! 353: 20 CONTINUE
! 354: CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
! 355: IF( LINFO.NE.0 )
! 356: $ GO TO 70
! 357: CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
! 358: IF( LINFO.NE.0 )
! 359: $ GO TO 70
! 360: *
! 361: * Perform the swapping tentatively:
! 362: *
! 363: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
! 364: $ WORK, M )
! 365: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S,
! 366: $ LDST )
! 367: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
! 368: $ WORK, M )
! 369: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T,
! 370: $ LDST )
! 371: CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
! 372: CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
! 373: CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
! 374: CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
! 375: *
! 376: * Triangularize the B-part by an RQ factorization.
! 377: * Apply transformation (from left) to A-part, giving S.
! 378: *
! 379: CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
! 380: IF( LINFO.NE.0 )
! 381: $ GO TO 70
! 382: CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK,
! 383: $ LINFO )
! 384: IF( LINFO.NE.0 )
! 385: $ GO TO 70
! 386: CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK,
! 387: $ LINFO )
! 388: IF( LINFO.NE.0 )
! 389: $ GO TO 70
! 390: *
! 391: * Compute F-norm(S21) in BRQA21. (T21 is 0.)
! 392: *
! 393: DSCALE = ZERO
! 394: DSUM = ONE
! 395: DO 30 I = 1, N2
! 396: CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
! 397: 30 CONTINUE
! 398: BRQA21 = DSCALE*SQRT( DSUM )
! 399: *
! 400: * Triangularize the B-part by a QR factorization.
! 401: * Apply transformation (from right) to A-part, giving S.
! 402: *
! 403: CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
! 404: IF( LINFO.NE.0 )
! 405: $ GO TO 70
! 406: CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST,
! 407: $ WORK, INFO )
! 408: CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST,
! 409: $ WORK, INFO )
! 410: IF( LINFO.NE.0 )
! 411: $ GO TO 70
! 412: *
! 413: * Compute F-norm(S21) in BQRA21. (T21 is 0.)
! 414: *
! 415: DSCALE = ZERO
! 416: DSUM = ONE
! 417: DO 40 I = 1, N2
! 418: CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
! 419: 40 CONTINUE
! 420: BQRA21 = DSCALE*SQRT( DSUM )
! 421: *
! 422: * Decide which method to use.
! 423: * Weak stability test:
! 424: * F-norm(S21) <= O(EPS * F-norm((S, T)))
! 425: *
! 426: IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN
! 427: CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
! 428: CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
! 429: CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
! 430: CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
! 431: ELSE IF( BRQA21.GE.THRESH ) THEN
! 432: GO TO 70
! 433: END IF
! 434: *
! 435: * Set lower triangle of B-part to zero
! 436: *
! 437: CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
! 438: *
! 439: IF( WANDS ) THEN
! 440: *
! 441: * Strong stability test:
! 442: * F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B)))
! 443: *
! 444: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
! 445: $ M )
! 446: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
! 447: $ WORK, M )
! 448: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
! 449: $ WORK( M*M+1 ), M )
! 450: DSCALE = ZERO
! 451: DSUM = ONE
! 452: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
! 453: *
! 454: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
! 455: $ M )
! 456: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
! 457: $ WORK, M )
! 458: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
! 459: $ WORK( M*M+1 ), M )
! 460: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
! 461: SS = DSCALE*SQRT( DSUM )
! 462: DTRONG = ( SS.LE.THRESH )
! 463: IF( .NOT.DTRONG )
! 464: $ GO TO 70
! 465: *
! 466: END IF
! 467: *
! 468: * If the swap is accepted ("weakly" and "strongly"), apply the
! 469: * transformations and set N1-by-N2 (2,1)-block to zero.
! 470: *
! 471: CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
! 472: *
! 473: * copy back M-by-M diagonal block starting at index J1 of (A, B)
! 474: *
! 475: CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
! 476: CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
! 477: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
! 478: *
! 479: * Standardize existing 2-by-2 blocks.
! 480: *
! 481: DO 50 I = 1, M*M
! 482: WORK(I) = ZERO
! 483: 50 CONTINUE
! 484: WORK( 1 ) = ONE
! 485: T( 1, 1 ) = ONE
! 486: IDUM = LWORK - M*M - 2
! 487: IF( N2.GT.1 ) THEN
! 488: CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE,
! 489: $ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
! 490: WORK( M+1 ) = -WORK( 2 )
! 491: WORK( M+2 ) = WORK( 1 )
! 492: T( N2, N2 ) = T( 1, 1 )
! 493: T( 1, 2 ) = -T( 2, 1 )
! 494: END IF
! 495: WORK( M*M ) = ONE
! 496: T( M, M ) = ONE
! 497: *
! 498: IF( N1.GT.1 ) THEN
! 499: CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB,
! 500: $ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
! 501: $ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
! 502: $ T( M, M-1 ) )
! 503: WORK( M*M ) = WORK( N2*M+N2+1 )
! 504: WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
! 505: T( M, M ) = T( N2+1, N2+1 )
! 506: T( M-1, M ) = -T( M, M-1 )
! 507: END IF
! 508: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ),
! 509: $ LDA, ZERO, WORK( M*M+1 ), N2 )
! 510: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ),
! 511: $ LDA )
! 512: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ),
! 513: $ LDB, ZERO, WORK( M*M+1 ), N2 )
! 514: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ),
! 515: $ LDB )
! 516: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
! 517: $ WORK( M*M+1 ), M )
! 518: CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
! 519: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
! 520: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
! 521: CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
! 522: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
! 523: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
! 524: CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
! 525: CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
! 526: $ WORK, M )
! 527: CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
! 528: *
! 529: * Accumulate transformations into Q and Z if requested.
! 530: *
! 531: IF( WANTQ ) THEN
! 532: CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
! 533: $ LDST, ZERO, WORK, N )
! 534: CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
! 535: *
! 536: END IF
! 537: *
! 538: IF( WANTZ ) THEN
! 539: CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
! 540: $ LDST, ZERO, WORK, N )
! 541: CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
! 542: *
! 543: END IF
! 544: *
! 545: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
! 546: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
! 547: *
! 548: I = J1 + M
! 549: IF( I.LE.N ) THEN
! 550: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
! 551: $ A( J1, I ), LDA, ZERO, WORK, M )
! 552: CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
! 553: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
! 554: $ B( J1, I ), LDA, ZERO, WORK, M )
! 555: CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
! 556: END IF
! 557: I = J1 - 1
! 558: IF( I.GT.0 ) THEN
! 559: CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
! 560: $ LDST, ZERO, WORK, I )
! 561: CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
! 562: CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
! 563: $ LDST, ZERO, WORK, I )
! 564: CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
! 565: END IF
! 566: *
! 567: * Exit with INFO = 0 if swap was successfully performed.
! 568: *
! 569: RETURN
! 570: *
! 571: END IF
! 572: *
! 573: * Exit with INFO = 1 if swap was rejected.
! 574: *
! 575: 70 CONTINUE
! 576: *
! 577: INFO = 1
! 578: RETURN
! 579: *
! 580: * End of DTGEX2
! 581: *
! 582: END
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