Annotation of rpl/lapack/lapack/ztgex2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
! 2: $ LDZ, J1, 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, N
! 12: * ..
! 13: * .. Array Arguments ..
! 14: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 15: $ Z( LDZ, * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)
! 22: * in an upper triangular matrix pair (A, B) by an unitary equivalence
! 23: * transformation.
! 24: *
! 25: * (A, B) must be in generalized Schur canonical form, that is, A and
! 26: * B are both upper triangular.
! 27: *
! 28: * Optionally, the matrices Q and Z of generalized Schur vectors are
! 29: * updated.
! 30: *
! 31: * Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
! 32: * Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
! 33: *
! 34: *
! 35: * Arguments
! 36: * =========
! 37: *
! 38: * WANTQ (input) LOGICAL
! 39: * .TRUE. : update the left transformation matrix Q;
! 40: * .FALSE.: do not update Q.
! 41: *
! 42: * WANTZ (input) LOGICAL
! 43: * .TRUE. : update the right transformation matrix Z;
! 44: * .FALSE.: do not update Z.
! 45: *
! 46: * N (input) INTEGER
! 47: * The order of the matrices A and B. N >= 0.
! 48: *
! 49: * A (input/output) COMPLEX*16 arrays, dimensions (LDA,N)
! 50: * On entry, the matrix A in the pair (A, B).
! 51: * On exit, the updated matrix A.
! 52: *
! 53: * LDA (input) INTEGER
! 54: * The leading dimension of the array A. LDA >= max(1,N).
! 55: *
! 56: * B (input/output) COMPLEX*16 arrays, dimensions (LDB,N)
! 57: * On entry, the matrix B in the pair (A, B).
! 58: * On exit, the updated matrix B.
! 59: *
! 60: * LDB (input) INTEGER
! 61: * The leading dimension of the array B. LDB >= max(1,N).
! 62: *
! 63: * Q (input/output) COMPLEX*16 array, dimension (LDZ,N)
! 64: * If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
! 65: * the updated matrix Q.
! 66: * Not referenced if WANTQ = .FALSE..
! 67: *
! 68: * LDQ (input) INTEGER
! 69: * The leading dimension of the array Q. LDQ >= 1;
! 70: * If WANTQ = .TRUE., LDQ >= N.
! 71: *
! 72: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
! 73: * If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,
! 74: * the updated matrix Z.
! 75: * Not referenced if WANTZ = .FALSE..
! 76: *
! 77: * LDZ (input) INTEGER
! 78: * The leading dimension of the array Z. LDZ >= 1;
! 79: * If WANTZ = .TRUE., LDZ >= N.
! 80: *
! 81: * J1 (input) INTEGER
! 82: * The index to the first block (A11, B11).
! 83: *
! 84: * INFO (output) INTEGER
! 85: * =0: Successful exit.
! 86: * =1: The transformed matrix pair (A, B) would be too far
! 87: * from generalized Schur form; the problem is ill-
! 88: * conditioned.
! 89: *
! 90: *
! 91: * Further Details
! 92: * ===============
! 93: *
! 94: * Based on contributions by
! 95: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 96: * Umea University, S-901 87 Umea, Sweden.
! 97: *
! 98: * In the current code both weak and strong stability tests are
! 99: * performed. The user can omit the strong stability test by changing
! 100: * the internal logical parameter WANDS to .FALSE.. See ref. [2] for
! 101: * details.
! 102: *
! 103: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
! 104: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
! 105: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
! 106: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
! 107: *
! 108: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
! 109: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
! 110: * Estimation: Theory, Algorithms and Software, Report UMINF-94.04,
! 111: * Department of Computing Science, Umea University, S-901 87 Umea,
! 112: * Sweden, 1994. Also as LAPACK Working Note 87. To appear in
! 113: * Numerical Algorithms, 1996.
! 114: *
! 115: * =====================================================================
! 116: *
! 117: * .. Parameters ..
! 118: COMPLEX*16 CZERO, CONE
! 119: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
! 120: $ CONE = ( 1.0D+0, 0.0D+0 ) )
! 121: DOUBLE PRECISION TEN
! 122: PARAMETER ( TEN = 10.0D+0 )
! 123: INTEGER LDST
! 124: PARAMETER ( LDST = 2 )
! 125: LOGICAL WANDS
! 126: PARAMETER ( WANDS = .TRUE. )
! 127: * ..
! 128: * .. Local Scalars ..
! 129: LOGICAL DTRONG, WEAK
! 130: INTEGER I, M
! 131: DOUBLE PRECISION CQ, CZ, EPS, SA, SB, SCALE, SMLNUM, SS, SUM,
! 132: $ THRESH, WS
! 133: COMPLEX*16 CDUM, F, G, SQ, SZ
! 134: * ..
! 135: * .. Local Arrays ..
! 136: COMPLEX*16 S( LDST, LDST ), T( LDST, LDST ), WORK( 8 )
! 137: * ..
! 138: * .. External Functions ..
! 139: DOUBLE PRECISION DLAMCH
! 140: EXTERNAL DLAMCH
! 141: * ..
! 142: * .. External Subroutines ..
! 143: EXTERNAL ZLACPY, ZLARTG, ZLASSQ, ZROT
! 144: * ..
! 145: * .. Intrinsic Functions ..
! 146: INTRINSIC ABS, DBLE, DCONJG, MAX, SQRT
! 147: * ..
! 148: * .. Executable Statements ..
! 149: *
! 150: INFO = 0
! 151: *
! 152: * Quick return if possible
! 153: *
! 154: IF( N.LE.1 )
! 155: $ RETURN
! 156: *
! 157: M = LDST
! 158: WEAK = .FALSE.
! 159: DTRONG = .FALSE.
! 160: *
! 161: * Make a local copy of selected block in (A, B)
! 162: *
! 163: CALL ZLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
! 164: CALL ZLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
! 165: *
! 166: * Compute the threshold for testing the acceptance of swapping.
! 167: *
! 168: EPS = DLAMCH( 'P' )
! 169: SMLNUM = DLAMCH( 'S' ) / EPS
! 170: SCALE = DBLE( CZERO )
! 171: SUM = DBLE( CONE )
! 172: CALL ZLACPY( 'Full', M, M, S, LDST, WORK, M )
! 173: CALL ZLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
! 174: CALL ZLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
! 175: SA = SCALE*SQRT( SUM )
! 176: THRESH = MAX( TEN*EPS*SA, SMLNUM )
! 177: *
! 178: * Compute unitary QL and RQ that swap 1-by-1 and 1-by-1 blocks
! 179: * using Givens rotations and perform the swap tentatively.
! 180: *
! 181: F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
! 182: G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
! 183: SA = ABS( S( 2, 2 ) )
! 184: SB = ABS( T( 2, 2 ) )
! 185: CALL ZLARTG( G, F, CZ, SZ, CDUM )
! 186: SZ = -SZ
! 187: CALL ZROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, CZ, DCONJG( SZ ) )
! 188: CALL ZROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, CZ, DCONJG( SZ ) )
! 189: IF( SA.GE.SB ) THEN
! 190: CALL ZLARTG( S( 1, 1 ), S( 2, 1 ), CQ, SQ, CDUM )
! 191: ELSE
! 192: CALL ZLARTG( T( 1, 1 ), T( 2, 1 ), CQ, SQ, CDUM )
! 193: END IF
! 194: CALL ZROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, CQ, SQ )
! 195: CALL ZROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, CQ, SQ )
! 196: *
! 197: * Weak stability test: |S21| + |T21| <= O(EPS F-norm((S, T)))
! 198: *
! 199: WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
! 200: WEAK = WS.LE.THRESH
! 201: IF( .NOT.WEAK )
! 202: $ GO TO 20
! 203: *
! 204: IF( WANDS ) THEN
! 205: *
! 206: * Strong stability test:
! 207: * F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A, B)))
! 208: *
! 209: CALL ZLACPY( 'Full', M, M, S, LDST, WORK, M )
! 210: CALL ZLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
! 211: CALL ZROT( 2, WORK, 1, WORK( 3 ), 1, CZ, -DCONJG( SZ ) )
! 212: CALL ZROT( 2, WORK( 5 ), 1, WORK( 7 ), 1, CZ, -DCONJG( SZ ) )
! 213: CALL ZROT( 2, WORK, 2, WORK( 2 ), 2, CQ, -SQ )
! 214: CALL ZROT( 2, WORK( 5 ), 2, WORK( 6 ), 2, CQ, -SQ )
! 215: DO 10 I = 1, 2
! 216: WORK( I ) = WORK( I ) - A( J1+I-1, J1 )
! 217: WORK( I+2 ) = WORK( I+2 ) - A( J1+I-1, J1+1 )
! 218: WORK( I+4 ) = WORK( I+4 ) - B( J1+I-1, J1 )
! 219: WORK( I+6 ) = WORK( I+6 ) - B( J1+I-1, J1+1 )
! 220: 10 CONTINUE
! 221: SCALE = DBLE( CZERO )
! 222: SUM = DBLE( CONE )
! 223: CALL ZLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
! 224: SS = SCALE*SQRT( SUM )
! 225: DTRONG = SS.LE.THRESH
! 226: IF( .NOT.DTRONG )
! 227: $ GO TO 20
! 228: END IF
! 229: *
! 230: * If the swap is accepted ("weakly" and "strongly"), apply the
! 231: * equivalence transformations to the original matrix pair (A,B)
! 232: *
! 233: CALL ZROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, CZ,
! 234: $ DCONJG( SZ ) )
! 235: CALL ZROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, CZ,
! 236: $ DCONJG( SZ ) )
! 237: CALL ZROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA, CQ, SQ )
! 238: CALL ZROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB, CQ, SQ )
! 239: *
! 240: * Set N1 by N2 (2,1) blocks to 0
! 241: *
! 242: A( J1+1, J1 ) = CZERO
! 243: B( J1+1, J1 ) = CZERO
! 244: *
! 245: * Accumulate transformations into Q and Z if requested.
! 246: *
! 247: IF( WANTZ )
! 248: $ CALL ZROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, CZ,
! 249: $ DCONJG( SZ ) )
! 250: IF( WANTQ )
! 251: $ CALL ZROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, CQ,
! 252: $ DCONJG( SQ ) )
! 253: *
! 254: * Exit with INFO = 0 if swap was successfully performed.
! 255: *
! 256: RETURN
! 257: *
! 258: * Exit with INFO = 1 if swap was rejected.
! 259: *
! 260: 20 CONTINUE
! 261: INFO = 1
! 262: RETURN
! 263: *
! 264: * End of ZTGEX2
! 265: *
! 266: END
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