1*> \brief \b ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZLAED7 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaed7.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaed7.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaed7.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, 22* LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, 23* GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, 24* INFO ) 25* 26* .. Scalar Arguments .. 27* INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, 28* $ TLVLS 29* DOUBLE PRECISION RHO 30* .. 31* .. Array Arguments .. 32* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), 33* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) 34* DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * ) 35* COMPLEX*16 Q( LDQ, * ), WORK( * ) 36* .. 37* 38* 39*> \par Purpose: 40* ============= 41*> 42*> \verbatim 43*> 44*> ZLAED7 computes the updated eigensystem of a diagonal 45*> matrix after modification by a rank-one symmetric matrix. This 46*> routine is used only for the eigenproblem which requires all 47*> eigenvalues and optionally eigenvectors of a dense or banded 48*> Hermitian matrix that has been reduced to tridiagonal form. 49*> 50*> T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out) 51*> 52*> where Z = Q**Hu, u is a vector of length N with ones in the 53*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. 54*> 55*> The eigenvectors of the original matrix are stored in Q, and the 56*> eigenvalues are in D. The algorithm consists of three stages: 57*> 58*> The first stage consists of deflating the size of the problem 59*> when there are multiple eigenvalues or if there is a zero in 60*> the Z vector. For each such occurrence the dimension of the 61*> secular equation problem is reduced by one. This stage is 62*> performed by the routine DLAED2. 63*> 64*> The second stage consists of calculating the updated 65*> eigenvalues. This is done by finding the roots of the secular 66*> equation via the routine DLAED4 (as called by SLAED3). 67*> This routine also calculates the eigenvectors of the current 68*> problem. 69*> 70*> The final stage consists of computing the updated eigenvectors 71*> directly using the updated eigenvalues. The eigenvectors for 72*> the current problem are multiplied with the eigenvectors from 73*> the overall problem. 74*> \endverbatim 75* 76* Arguments: 77* ========== 78* 79*> \param[in] N 80*> \verbatim 81*> N is INTEGER 82*> The dimension of the symmetric tridiagonal matrix. N >= 0. 83*> \endverbatim 84*> 85*> \param[in] CUTPNT 86*> \verbatim 87*> CUTPNT is INTEGER 88*> Contains the location of the last eigenvalue in the leading 89*> sub-matrix. min(1,N) <= CUTPNT <= N. 90*> \endverbatim 91*> 92*> \param[in] QSIZ 93*> \verbatim 94*> QSIZ is INTEGER 95*> The dimension of the unitary matrix used to reduce 96*> the full matrix to tridiagonal form. QSIZ >= N. 97*> \endverbatim 98*> 99*> \param[in] TLVLS 100*> \verbatim 101*> TLVLS is INTEGER 102*> The total number of merging levels in the overall divide and 103*> conquer tree. 104*> \endverbatim 105*> 106*> \param[in] CURLVL 107*> \verbatim 108*> CURLVL is INTEGER 109*> The current level in the overall merge routine, 110*> 0 <= curlvl <= tlvls. 111*> \endverbatim 112*> 113*> \param[in] CURPBM 114*> \verbatim 115*> CURPBM is INTEGER 116*> The current problem in the current level in the overall 117*> merge routine (counting from upper left to lower right). 118*> \endverbatim 119*> 120*> \param[in,out] D 121*> \verbatim 122*> D is DOUBLE PRECISION array, dimension (N) 123*> On entry, the eigenvalues of the rank-1-perturbed matrix. 124*> On exit, the eigenvalues of the repaired matrix. 125*> \endverbatim 126*> 127*> \param[in,out] Q 128*> \verbatim 129*> Q is COMPLEX*16 array, dimension (LDQ,N) 130*> On entry, the eigenvectors of the rank-1-perturbed matrix. 131*> On exit, the eigenvectors of the repaired tridiagonal matrix. 132*> \endverbatim 133*> 134*> \param[in] LDQ 135*> \verbatim 136*> LDQ is INTEGER 137*> The leading dimension of the array Q. LDQ >= max(1,N). 138*> \endverbatim 139*> 140*> \param[in] RHO 141*> \verbatim 142*> RHO is DOUBLE PRECISION 143*> Contains the subdiagonal element used to create the rank-1 144*> modification. 145*> \endverbatim 146*> 147*> \param[out] INDXQ 148*> \verbatim 149*> INDXQ is INTEGER array, dimension (N) 150*> This contains the permutation which will reintegrate the 151*> subproblem just solved back into sorted order, 152*> ie. D( INDXQ( I = 1, N ) ) will be in ascending order. 153*> \endverbatim 154*> 155*> \param[out] IWORK 156*> \verbatim 157*> IWORK is INTEGER array, dimension (4*N) 158*> \endverbatim 159*> 160*> \param[out] RWORK 161*> \verbatim 162*> RWORK is DOUBLE PRECISION array, 163*> dimension (3*N+2*QSIZ*N) 164*> \endverbatim 165*> 166*> \param[out] WORK 167*> \verbatim 168*> WORK is COMPLEX*16 array, dimension (QSIZ*N) 169*> \endverbatim 170*> 171*> \param[in,out] QSTORE 172*> \verbatim 173*> QSTORE is DOUBLE PRECISION array, dimension (N**2+1) 174*> Stores eigenvectors of submatrices encountered during 175*> divide and conquer, packed together. QPTR points to 176*> beginning of the submatrices. 177*> \endverbatim 178*> 179*> \param[in,out] QPTR 180*> \verbatim 181*> QPTR is INTEGER array, dimension (N+2) 182*> List of indices pointing to beginning of submatrices stored 183*> in QSTORE. The submatrices are numbered starting at the 184*> bottom left of the divide and conquer tree, from left to 185*> right and bottom to top. 186*> \endverbatim 187*> 188*> \param[in] PRMPTR 189*> \verbatim 190*> PRMPTR is INTEGER array, dimension (N lg N) 191*> Contains a list of pointers which indicate where in PERM a 192*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) 193*> indicates the size of the permutation and also the size of 194*> the full, non-deflated problem. 195*> \endverbatim 196*> 197*> \param[in] PERM 198*> \verbatim 199*> PERM is INTEGER array, dimension (N lg N) 200*> Contains the permutations (from deflation and sorting) to be 201*> applied to each eigenblock. 202*> \endverbatim 203*> 204*> \param[in] GIVPTR 205*> \verbatim 206*> GIVPTR is INTEGER array, dimension (N lg N) 207*> Contains a list of pointers which indicate where in GIVCOL a 208*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) 209*> indicates the number of Givens rotations. 210*> \endverbatim 211*> 212*> \param[in] GIVCOL 213*> \verbatim 214*> GIVCOL is INTEGER array, dimension (2, N lg N) 215*> Each pair of numbers indicates a pair of columns to take place 216*> in a Givens rotation. 217*> \endverbatim 218*> 219*> \param[in] GIVNUM 220*> \verbatim 221*> GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) 222*> Each number indicates the S value to be used in the 223*> corresponding Givens rotation. 224*> \endverbatim 225*> 226*> \param[out] INFO 227*> \verbatim 228*> INFO is INTEGER 229*> = 0: successful exit. 230*> < 0: if INFO = -i, the i-th argument had an illegal value. 231*> > 0: if INFO = 1, an eigenvalue did not converge 232*> \endverbatim 233* 234* Authors: 235* ======== 236* 237*> \author Univ. of Tennessee 238*> \author Univ. of California Berkeley 239*> \author Univ. of Colorado Denver 240*> \author NAG Ltd. 241* 242*> \ingroup complex16OTHERcomputational 243* 244* ===================================================================== 245 SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, 246 $ LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, 247 $ GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, 248 $ INFO ) 249* 250* -- LAPACK computational routine -- 251* -- LAPACK is a software package provided by Univ. of Tennessee, -- 252* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 253* 254* .. Scalar Arguments .. 255 INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, 256 $ TLVLS 257 DOUBLE PRECISION RHO 258* .. 259* .. Array Arguments .. 260 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), 261 $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) 262 DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * ) 263 COMPLEX*16 Q( LDQ, * ), WORK( * ) 264* .. 265* 266* ===================================================================== 267* 268* .. Local Scalars .. 269 INTEGER COLTYP, CURR, I, IDLMDA, INDX, 270 $ INDXC, INDXP, IQ, IW, IZ, K, N1, N2, PTR 271* .. 272* .. External Subroutines .. 273 EXTERNAL DLAED9, DLAEDA, DLAMRG, XERBLA, ZLACRM, ZLAED8 274* .. 275* .. Intrinsic Functions .. 276 INTRINSIC MAX, MIN 277* .. 278* .. Executable Statements .. 279* 280* Test the input parameters. 281* 282 INFO = 0 283* 284* IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN 285* INFO = -1 286* ELSE IF( N.LT.0 ) THEN 287 IF( N.LT.0 ) THEN 288 INFO = -1 289 ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN 290 INFO = -2 291 ELSE IF( QSIZ.LT.N ) THEN 292 INFO = -3 293 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN 294 INFO = -9 295 END IF 296 IF( INFO.NE.0 ) THEN 297 CALL XERBLA( 'ZLAED7', -INFO ) 298 RETURN 299 END IF 300* 301* Quick return if possible 302* 303 IF( N.EQ.0 ) 304 $ RETURN 305* 306* The following values are for bookkeeping purposes only. They are 307* integer pointers which indicate the portion of the workspace 308* used by a particular array in DLAED2 and SLAED3. 309* 310 IZ = 1 311 IDLMDA = IZ + N 312 IW = IDLMDA + N 313 IQ = IW + N 314* 315 INDX = 1 316 INDXC = INDX + N 317 COLTYP = INDXC + N 318 INDXP = COLTYP + N 319* 320* Form the z-vector which consists of the last row of Q_1 and the 321* first row of Q_2. 322* 323 PTR = 1 + 2**TLVLS 324 DO 10 I = 1, CURLVL - 1 325 PTR = PTR + 2**( TLVLS-I ) 326 10 CONTINUE 327 CURR = PTR + CURPBM 328 CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, 329 $ GIVCOL, GIVNUM, QSTORE, QPTR, RWORK( IZ ), 330 $ RWORK( IZ+N ), INFO ) 331* 332* When solving the final problem, we no longer need the stored data, 333* so we will overwrite the data from this level onto the previously 334* used storage space. 335* 336 IF( CURLVL.EQ.TLVLS ) THEN 337 QPTR( CURR ) = 1 338 PRMPTR( CURR ) = 1 339 GIVPTR( CURR ) = 1 340 END IF 341* 342* Sort and Deflate eigenvalues. 343* 344 CALL ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, RWORK( IZ ), 345 $ RWORK( IDLMDA ), WORK, QSIZ, RWORK( IW ), 346 $ IWORK( INDXP ), IWORK( INDX ), INDXQ, 347 $ PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ), 348 $ GIVCOL( 1, GIVPTR( CURR ) ), 349 $ GIVNUM( 1, GIVPTR( CURR ) ), INFO ) 350 PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N 351 GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR ) 352* 353* Solve Secular Equation. 354* 355 IF( K.NE.0 ) THEN 356 CALL DLAED9( K, 1, K, N, D, RWORK( IQ ), K, RHO, 357 $ RWORK( IDLMDA ), RWORK( IW ), 358 $ QSTORE( QPTR( CURR ) ), K, INFO ) 359 CALL ZLACRM( QSIZ, K, WORK, QSIZ, QSTORE( QPTR( CURR ) ), K, Q, 360 $ LDQ, RWORK( IQ ) ) 361 QPTR( CURR+1 ) = QPTR( CURR ) + K**2 362 IF( INFO.NE.0 ) THEN 363 RETURN 364 END IF 365* 366* Prepare the INDXQ sorting premutation. 367* 368 N1 = K 369 N2 = N - K 370 CALL DLAMRG( N1, N2, D, 1, -1, INDXQ ) 371 ELSE 372 QPTR( CURR+1 ) = QPTR( CURR ) 373 DO 20 I = 1, N 374 INDXQ( I ) = I 375 20 CONTINUE 376 END IF 377* 378 RETURN 379* 380* End of ZLAED7 381* 382 END 383