Annotation of rpl/lapack/lapack/zlaed8.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
! 2: $ Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
! 3: $ GIVCOL, GIVNUM, INFO )
! 4: *
! 5: * -- LAPACK routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
! 12: DOUBLE PRECISION RHO
! 13: * ..
! 14: * .. Array Arguments ..
! 15: INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
! 16: $ INDXQ( * ), PERM( * )
! 17: DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
! 18: $ Z( * )
! 19: COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * ZLAED8 merges the two sets of eigenvalues together into a single
! 26: * sorted set. Then it tries to deflate the size of the problem.
! 27: * There are two ways in which deflation can occur: when two or more
! 28: * eigenvalues are close together or if there is a tiny element in the
! 29: * Z vector. For each such occurrence the order of the related secular
! 30: * equation problem is reduced by one.
! 31: *
! 32: * Arguments
! 33: * =========
! 34: *
! 35: * K (output) INTEGER
! 36: * Contains the number of non-deflated eigenvalues.
! 37: * This is the order of the related secular equation.
! 38: *
! 39: * N (input) INTEGER
! 40: * The dimension of the symmetric tridiagonal matrix. N >= 0.
! 41: *
! 42: * QSIZ (input) INTEGER
! 43: * The dimension of the unitary matrix used to reduce
! 44: * the dense or band matrix to tridiagonal form.
! 45: * QSIZ >= N if ICOMPQ = 1.
! 46: *
! 47: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
! 48: * On entry, Q contains the eigenvectors of the partially solved
! 49: * system which has been previously updated in matrix
! 50: * multiplies with other partially solved eigensystems.
! 51: * On exit, Q contains the trailing (N-K) updated eigenvectors
! 52: * (those which were deflated) in its last N-K columns.
! 53: *
! 54: * LDQ (input) INTEGER
! 55: * The leading dimension of the array Q. LDQ >= max( 1, N ).
! 56: *
! 57: * D (input/output) DOUBLE PRECISION array, dimension (N)
! 58: * On entry, D contains the eigenvalues of the two submatrices to
! 59: * be combined. On exit, D contains the trailing (N-K) updated
! 60: * eigenvalues (those which were deflated) sorted into increasing
! 61: * order.
! 62: *
! 63: * RHO (input/output) DOUBLE PRECISION
! 64: * Contains the off diagonal element associated with the rank-1
! 65: * cut which originally split the two submatrices which are now
! 66: * being recombined. RHO is modified during the computation to
! 67: * the value required by DLAED3.
! 68: *
! 69: * CUTPNT (input) INTEGER
! 70: * Contains the location of the last eigenvalue in the leading
! 71: * sub-matrix. MIN(1,N) <= CUTPNT <= N.
! 72: *
! 73: * Z (input) DOUBLE PRECISION array, dimension (N)
! 74: * On input this vector contains the updating vector (the last
! 75: * row of the first sub-eigenvector matrix and the first row of
! 76: * the second sub-eigenvector matrix). The contents of Z are
! 77: * destroyed during the updating process.
! 78: *
! 79: * DLAMDA (output) DOUBLE PRECISION array, dimension (N)
! 80: * Contains a copy of the first K eigenvalues which will be used
! 81: * by DLAED3 to form the secular equation.
! 82: *
! 83: * Q2 (output) COMPLEX*16 array, dimension (LDQ2,N)
! 84: * If ICOMPQ = 0, Q2 is not referenced. Otherwise,
! 85: * Contains a copy of the first K eigenvectors which will be used
! 86: * by DLAED7 in a matrix multiply (DGEMM) to update the new
! 87: * eigenvectors.
! 88: *
! 89: * LDQ2 (input) INTEGER
! 90: * The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
! 91: *
! 92: * W (output) DOUBLE PRECISION array, dimension (N)
! 93: * This will hold the first k values of the final
! 94: * deflation-altered z-vector and will be passed to DLAED3.
! 95: *
! 96: * INDXP (workspace) INTEGER array, dimension (N)
! 97: * This will contain the permutation used to place deflated
! 98: * values of D at the end of the array. On output INDXP(1:K)
! 99: * points to the nondeflated D-values and INDXP(K+1:N)
! 100: * points to the deflated eigenvalues.
! 101: *
! 102: * INDX (workspace) INTEGER array, dimension (N)
! 103: * This will contain the permutation used to sort the contents of
! 104: * D into ascending order.
! 105: *
! 106: * INDXQ (input) INTEGER array, dimension (N)
! 107: * This contains the permutation which separately sorts the two
! 108: * sub-problems in D into ascending order. Note that elements in
! 109: * the second half of this permutation must first have CUTPNT
! 110: * added to their values in order to be accurate.
! 111: *
! 112: * PERM (output) INTEGER array, dimension (N)
! 113: * Contains the permutations (from deflation and sorting) to be
! 114: * applied to each eigenblock.
! 115: *
! 116: * GIVPTR (output) INTEGER
! 117: * Contains the number of Givens rotations which took place in
! 118: * this subproblem.
! 119: *
! 120: * GIVCOL (output) INTEGER array, dimension (2, N)
! 121: * Each pair of numbers indicates a pair of columns to take place
! 122: * in a Givens rotation.
! 123: *
! 124: * GIVNUM (output) DOUBLE PRECISION array, dimension (2, N)
! 125: * Each number indicates the S value to be used in the
! 126: * corresponding Givens rotation.
! 127: *
! 128: * INFO (output) INTEGER
! 129: * = 0: successful exit.
! 130: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 131: *
! 132: * =====================================================================
! 133: *
! 134: * .. Parameters ..
! 135: DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
! 136: PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
! 137: $ TWO = 2.0D0, EIGHT = 8.0D0 )
! 138: * ..
! 139: * .. Local Scalars ..
! 140: INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
! 141: DOUBLE PRECISION C, EPS, S, T, TAU, TOL
! 142: * ..
! 143: * .. External Functions ..
! 144: INTEGER IDAMAX
! 145: DOUBLE PRECISION DLAMCH, DLAPY2
! 146: EXTERNAL IDAMAX, DLAMCH, DLAPY2
! 147: * ..
! 148: * .. External Subroutines ..
! 149: EXTERNAL DCOPY, DLAMRG, DSCAL, XERBLA, ZCOPY, ZDROT,
! 150: $ ZLACPY
! 151: * ..
! 152: * .. Intrinsic Functions ..
! 153: INTRINSIC ABS, MAX, MIN, SQRT
! 154: * ..
! 155: * .. Executable Statements ..
! 156: *
! 157: * Test the input parameters.
! 158: *
! 159: INFO = 0
! 160: *
! 161: IF( N.LT.0 ) THEN
! 162: INFO = -2
! 163: ELSE IF( QSIZ.LT.N ) THEN
! 164: INFO = -3
! 165: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
! 166: INFO = -5
! 167: ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
! 168: INFO = -8
! 169: ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
! 170: INFO = -12
! 171: END IF
! 172: IF( INFO.NE.0 ) THEN
! 173: CALL XERBLA( 'ZLAED8', -INFO )
! 174: RETURN
! 175: END IF
! 176: *
! 177: * Quick return if possible
! 178: *
! 179: IF( N.EQ.0 )
! 180: $ RETURN
! 181: *
! 182: N1 = CUTPNT
! 183: N2 = N - N1
! 184: N1P1 = N1 + 1
! 185: *
! 186: IF( RHO.LT.ZERO ) THEN
! 187: CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
! 188: END IF
! 189: *
! 190: * Normalize z so that norm(z) = 1
! 191: *
! 192: T = ONE / SQRT( TWO )
! 193: DO 10 J = 1, N
! 194: INDX( J ) = J
! 195: 10 CONTINUE
! 196: CALL DSCAL( N, T, Z, 1 )
! 197: RHO = ABS( TWO*RHO )
! 198: *
! 199: * Sort the eigenvalues into increasing order
! 200: *
! 201: DO 20 I = CUTPNT + 1, N
! 202: INDXQ( I ) = INDXQ( I ) + CUTPNT
! 203: 20 CONTINUE
! 204: DO 30 I = 1, N
! 205: DLAMDA( I ) = D( INDXQ( I ) )
! 206: W( I ) = Z( INDXQ( I ) )
! 207: 30 CONTINUE
! 208: I = 1
! 209: J = CUTPNT + 1
! 210: CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
! 211: DO 40 I = 1, N
! 212: D( I ) = DLAMDA( INDX( I ) )
! 213: Z( I ) = W( INDX( I ) )
! 214: 40 CONTINUE
! 215: *
! 216: * Calculate the allowable deflation tolerance
! 217: *
! 218: IMAX = IDAMAX( N, Z, 1 )
! 219: JMAX = IDAMAX( N, D, 1 )
! 220: EPS = DLAMCH( 'Epsilon' )
! 221: TOL = EIGHT*EPS*ABS( D( JMAX ) )
! 222: *
! 223: * If the rank-1 modifier is small enough, no more needs to be done
! 224: * -- except to reorganize Q so that its columns correspond with the
! 225: * elements in D.
! 226: *
! 227: IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
! 228: K = 0
! 229: DO 50 J = 1, N
! 230: PERM( J ) = INDXQ( INDX( J ) )
! 231: CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
! 232: 50 CONTINUE
! 233: CALL ZLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), LDQ )
! 234: RETURN
! 235: END IF
! 236: *
! 237: * If there are multiple eigenvalues then the problem deflates. Here
! 238: * the number of equal eigenvalues are found. As each equal
! 239: * eigenvalue is found, an elementary reflector is computed to rotate
! 240: * the corresponding eigensubspace so that the corresponding
! 241: * components of Z are zero in this new basis.
! 242: *
! 243: K = 0
! 244: GIVPTR = 0
! 245: K2 = N + 1
! 246: DO 60 J = 1, N
! 247: IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
! 248: *
! 249: * Deflate due to small z component.
! 250: *
! 251: K2 = K2 - 1
! 252: INDXP( K2 ) = J
! 253: IF( J.EQ.N )
! 254: $ GO TO 100
! 255: ELSE
! 256: JLAM = J
! 257: GO TO 70
! 258: END IF
! 259: 60 CONTINUE
! 260: 70 CONTINUE
! 261: J = J + 1
! 262: IF( J.GT.N )
! 263: $ GO TO 90
! 264: IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
! 265: *
! 266: * Deflate due to small z component.
! 267: *
! 268: K2 = K2 - 1
! 269: INDXP( K2 ) = J
! 270: ELSE
! 271: *
! 272: * Check if eigenvalues are close enough to allow deflation.
! 273: *
! 274: S = Z( JLAM )
! 275: C = Z( J )
! 276: *
! 277: * Find sqrt(a**2+b**2) without overflow or
! 278: * destructive underflow.
! 279: *
! 280: TAU = DLAPY2( C, S )
! 281: T = D( J ) - D( JLAM )
! 282: C = C / TAU
! 283: S = -S / TAU
! 284: IF( ABS( T*C*S ).LE.TOL ) THEN
! 285: *
! 286: * Deflation is possible.
! 287: *
! 288: Z( J ) = TAU
! 289: Z( JLAM ) = ZERO
! 290: *
! 291: * Record the appropriate Givens rotation
! 292: *
! 293: GIVPTR = GIVPTR + 1
! 294: GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
! 295: GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
! 296: GIVNUM( 1, GIVPTR ) = C
! 297: GIVNUM( 2, GIVPTR ) = S
! 298: CALL ZDROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
! 299: $ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
! 300: T = D( JLAM )*C*C + D( J )*S*S
! 301: D( J ) = D( JLAM )*S*S + D( J )*C*C
! 302: D( JLAM ) = T
! 303: K2 = K2 - 1
! 304: I = 1
! 305: 80 CONTINUE
! 306: IF( K2+I.LE.N ) THEN
! 307: IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
! 308: INDXP( K2+I-1 ) = INDXP( K2+I )
! 309: INDXP( K2+I ) = JLAM
! 310: I = I + 1
! 311: GO TO 80
! 312: ELSE
! 313: INDXP( K2+I-1 ) = JLAM
! 314: END IF
! 315: ELSE
! 316: INDXP( K2+I-1 ) = JLAM
! 317: END IF
! 318: JLAM = J
! 319: ELSE
! 320: K = K + 1
! 321: W( K ) = Z( JLAM )
! 322: DLAMDA( K ) = D( JLAM )
! 323: INDXP( K ) = JLAM
! 324: JLAM = J
! 325: END IF
! 326: END IF
! 327: GO TO 70
! 328: 90 CONTINUE
! 329: *
! 330: * Record the last eigenvalue.
! 331: *
! 332: K = K + 1
! 333: W( K ) = Z( JLAM )
! 334: DLAMDA( K ) = D( JLAM )
! 335: INDXP( K ) = JLAM
! 336: *
! 337: 100 CONTINUE
! 338: *
! 339: * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
! 340: * and Q2 respectively. The eigenvalues/vectors which were not
! 341: * deflated go into the first K slots of DLAMDA and Q2 respectively,
! 342: * while those which were deflated go into the last N - K slots.
! 343: *
! 344: DO 110 J = 1, N
! 345: JP = INDXP( J )
! 346: DLAMDA( J ) = D( JP )
! 347: PERM( J ) = INDXQ( INDX( JP ) )
! 348: CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
! 349: 110 CONTINUE
! 350: *
! 351: * The deflated eigenvalues and their corresponding vectors go back
! 352: * into the last N - K slots of D and Q respectively.
! 353: *
! 354: IF( K.LT.N ) THEN
! 355: CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
! 356: CALL ZLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ),
! 357: $ LDQ )
! 358: END IF
! 359: *
! 360: RETURN
! 361: *
! 362: * End of ZLAED8
! 363: *
! 364: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to zlaed8.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack