1*> \brief \b SLAED7 used by sstedc. 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 SLAED7 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaed7.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaed7.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed7.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, 22* LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, 23* PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, 24* INFO ) 25* 26* .. Scalar Arguments .. 27* INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, 28* $ QSIZ, TLVLS 29* REAL RHO 30* .. 31* .. Array Arguments .. 32* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), 33* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) 34* REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ), 35* $ QSTORE( * ), WORK( * ) 36* .. 37* 38* 39*> \par Purpose: 40* ============= 41*> 42*> \verbatim 43*> 44*> SLAED7 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 symmetric matrix 48*> that has been reduced to tridiagonal form. SLAED1 handles 49*> the case in which all eigenvalues and eigenvectors of a symmetric 50*> tridiagonal matrix are desired. 51*> 52*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) 53*> 54*> where Z = Q**Tu, u is a vector of length N with ones in the 55*> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. 56*> 57*> The eigenvectors of the original matrix are stored in Q, and the 58*> eigenvalues are in D. The algorithm consists of three stages: 59*> 60*> The first stage consists of deflating the size of the problem 61*> when there are multiple eigenvalues or if there is a zero in 62*> the Z vector. For each such occurrence the dimension of the 63*> secular equation problem is reduced by one. This stage is 64*> performed by the routine SLAED8. 65*> 66*> The second stage consists of calculating the updated 67*> eigenvalues. This is done by finding the roots of the secular 68*> equation via the routine SLAED4 (as called by SLAED9). 69*> This routine also calculates the eigenvectors of the current 70*> problem. 71*> 72*> The final stage consists of computing the updated eigenvectors 73*> directly using the updated eigenvalues. The eigenvectors for 74*> the current problem are multiplied with the eigenvectors from 75*> the overall problem. 76*> \endverbatim 77* 78* Arguments: 79* ========== 80* 81*> \param[in] ICOMPQ 82*> \verbatim 83*> ICOMPQ is INTEGER 84*> = 0: Compute eigenvalues only. 85*> = 1: Compute eigenvectors of original dense symmetric matrix 86*> also. On entry, Q contains the orthogonal matrix used 87*> to reduce the original matrix to tridiagonal form. 88*> \endverbatim 89*> 90*> \param[in] N 91*> \verbatim 92*> N is INTEGER 93*> The dimension of the symmetric tridiagonal matrix. N >= 0. 94*> \endverbatim 95*> 96*> \param[in] QSIZ 97*> \verbatim 98*> QSIZ is INTEGER 99*> The dimension of the orthogonal matrix used to reduce 100*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. 101*> \endverbatim 102*> 103*> \param[in] TLVLS 104*> \verbatim 105*> TLVLS is INTEGER 106*> The total number of merging levels in the overall divide and 107*> conquer tree. 108*> \endverbatim 109*> 110*> \param[in] CURLVL 111*> \verbatim 112*> CURLVL is INTEGER 113*> The current level in the overall merge routine, 114*> 0 <= CURLVL <= TLVLS. 115*> \endverbatim 116*> 117*> \param[in] CURPBM 118*> \verbatim 119*> CURPBM is INTEGER 120*> The current problem in the current level in the overall 121*> merge routine (counting from upper left to lower right). 122*> \endverbatim 123*> 124*> \param[in,out] D 125*> \verbatim 126*> D is REAL array, dimension (N) 127*> On entry, the eigenvalues of the rank-1-perturbed matrix. 128*> On exit, the eigenvalues of the repaired matrix. 129*> \endverbatim 130*> 131*> \param[in,out] Q 132*> \verbatim 133*> Q is REAL array, dimension (LDQ, N) 134*> On entry, the eigenvectors of the rank-1-perturbed matrix. 135*> On exit, the eigenvectors of the repaired tridiagonal matrix. 136*> \endverbatim 137*> 138*> \param[in] LDQ 139*> \verbatim 140*> LDQ is INTEGER 141*> The leading dimension of the array Q. LDQ >= max(1,N). 142*> \endverbatim 143*> 144*> \param[out] INDXQ 145*> \verbatim 146*> INDXQ is INTEGER array, dimension (N) 147*> The permutation which will reintegrate the subproblem just 148*> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) 149*> will be in ascending order. 150*> \endverbatim 151*> 152*> \param[in] RHO 153*> \verbatim 154*> RHO is REAL 155*> The subdiagonal element used to create the rank-1 156*> modification. 157*> \endverbatim 158*> 159*> \param[in] CUTPNT 160*> \verbatim 161*> CUTPNT is INTEGER 162*> Contains the location of the last eigenvalue in the leading 163*> sub-matrix. min(1,N) <= CUTPNT <= N. 164*> \endverbatim 165*> 166*> \param[in,out] QSTORE 167*> \verbatim 168*> QSTORE is REAL array, dimension (N**2+1) 169*> Stores eigenvectors of submatrices encountered during 170*> divide and conquer, packed together. QPTR points to 171*> beginning of the submatrices. 172*> \endverbatim 173*> 174*> \param[in,out] QPTR 175*> \verbatim 176*> QPTR is INTEGER array, dimension (N+2) 177*> List of indices pointing to beginning of submatrices stored 178*> in QSTORE. The submatrices are numbered starting at the 179*> bottom left of the divide and conquer tree, from left to 180*> right and bottom to top. 181*> \endverbatim 182*> 183*> \param[in] PRMPTR 184*> \verbatim 185*> PRMPTR is INTEGER array, dimension (N lg N) 186*> Contains a list of pointers which indicate where in PERM a 187*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) 188*> indicates the size of the permutation and also the size of 189*> the full, non-deflated problem. 190*> \endverbatim 191*> 192*> \param[in] PERM 193*> \verbatim 194*> PERM is INTEGER array, dimension (N lg N) 195*> Contains the permutations (from deflation and sorting) to be 196*> applied to each eigenblock. 197*> \endverbatim 198*> 199*> \param[in] GIVPTR 200*> \verbatim 201*> GIVPTR is INTEGER array, dimension (N lg N) 202*> Contains a list of pointers which indicate where in GIVCOL a 203*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) 204*> indicates the number of Givens rotations. 205*> \endverbatim 206*> 207*> \param[in] GIVCOL 208*> \verbatim 209*> GIVCOL is INTEGER array, dimension (2, N lg N) 210*> Each pair of numbers indicates a pair of columns to take place 211*> in a Givens rotation. 212*> \endverbatim 213*> 214*> \param[in] GIVNUM 215*> \verbatim 216*> GIVNUM is REAL array, dimension (2, N lg N) 217*> Each number indicates the S value to be used in the 218*> corresponding Givens rotation. 219*> \endverbatim 220*> 221*> \param[out] WORK 222*> \verbatim 223*> WORK is REAL array, dimension (3*N+2*QSIZ*N) 224*> \endverbatim 225*> 226*> \param[out] IWORK 227*> \verbatim 228*> IWORK is INTEGER array, dimension (4*N) 229*> \endverbatim 230*> 231*> \param[out] INFO 232*> \verbatim 233*> INFO is INTEGER 234*> = 0: successful exit. 235*> < 0: if INFO = -i, the i-th argument had an illegal value. 236*> > 0: if INFO = 1, an eigenvalue did not converge 237*> \endverbatim 238* 239* Authors: 240* ======== 241* 242*> \author Univ. of Tennessee 243*> \author Univ. of California Berkeley 244*> \author Univ. of Colorado Denver 245*> \author NAG Ltd. 246* 247*> \date June 2016 248* 249*> \ingroup auxOTHERcomputational 250* 251*> \par Contributors: 252* ================== 253*> 254*> Jeff Rutter, Computer Science Division, University of California 255*> at Berkeley, USA 256* 257* ===================================================================== 258 SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, 259 $ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, 260 $ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, 261 $ INFO ) 262* 263* -- LAPACK computational routine (version 3.7.0) -- 264* -- LAPACK is a software package provided by Univ. of Tennessee, -- 265* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 266* June 2016 267* 268* .. Scalar Arguments .. 269 INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, 270 $ QSIZ, TLVLS 271 REAL RHO 272* .. 273* .. Array Arguments .. 274 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), 275 $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) 276 REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ), 277 $ QSTORE( * ), WORK( * ) 278* .. 279* 280* ===================================================================== 281* 282* .. Parameters .. 283 REAL ONE, ZERO 284 PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0 ) 285* .. 286* .. Local Scalars .. 287 INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP, 288 $ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR 289* .. 290* .. External Subroutines .. 291 EXTERNAL SGEMM, SLAED8, SLAED9, SLAEDA, SLAMRG, XERBLA 292* .. 293* .. Intrinsic Functions .. 294 INTRINSIC MAX, MIN 295* .. 296* .. Executable Statements .. 297* 298* Test the input parameters. 299* 300 INFO = 0 301* 302 IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN 303 INFO = -1 304 ELSE IF( N.LT.0 ) THEN 305 INFO = -2 306 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN 307 INFO = -3 308 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN 309 INFO = -9 310 ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN 311 INFO = -12 312 END IF 313 IF( INFO.NE.0 ) THEN 314 CALL XERBLA( 'SLAED7', -INFO ) 315 RETURN 316 END IF 317* 318* Quick return if possible 319* 320 IF( N.EQ.0 ) 321 $ RETURN 322* 323* The following values are for bookkeeping purposes only. They are 324* integer pointers which indicate the portion of the workspace 325* used by a particular array in SLAED8 and SLAED9. 326* 327 IF( ICOMPQ.EQ.1 ) THEN 328 LDQ2 = QSIZ 329 ELSE 330 LDQ2 = N 331 END IF 332* 333 IZ = 1 334 IDLMDA = IZ + N 335 IW = IDLMDA + N 336 IQ2 = IW + N 337 IS = IQ2 + N*LDQ2 338* 339 INDX = 1 340 INDXC = INDX + N 341 COLTYP = INDXC + N 342 INDXP = COLTYP + N 343* 344* Form the z-vector which consists of the last row of Q_1 and the 345* first row of Q_2. 346* 347 PTR = 1 + 2**TLVLS 348 DO 10 I = 1, CURLVL - 1 349 PTR = PTR + 2**( TLVLS-I ) 350 10 CONTINUE 351 CURR = PTR + CURPBM 352 CALL SLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, 353 $ GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ), 354 $ WORK( IZ+N ), INFO ) 355* 356* When solving the final problem, we no longer need the stored data, 357* so we will overwrite the data from this level onto the previously 358* used storage space. 359* 360 IF( CURLVL.EQ.TLVLS ) THEN 361 QPTR( CURR ) = 1 362 PRMPTR( CURR ) = 1 363 GIVPTR( CURR ) = 1 364 END IF 365* 366* Sort and Deflate eigenvalues. 367* 368 CALL SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, 369 $ WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2, 370 $ WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ), 371 $ GIVCOL( 1, GIVPTR( CURR ) ), 372 $ GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ), 373 $ IWORK( INDX ), INFO ) 374 PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N 375 GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR ) 376* 377* Solve Secular Equation. 378* 379 IF( K.NE.0 ) THEN 380 CALL SLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ), 381 $ WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO ) 382 IF( INFO.NE.0 ) 383 $ GO TO 30 384 IF( ICOMPQ.EQ.1 ) THEN 385 CALL SGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2, 386 $ QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ ) 387 END IF 388 QPTR( CURR+1 ) = QPTR( CURR ) + K**2 389* 390* Prepare the INDXQ sorting permutation. 391* 392 N1 = K 393 N2 = N - K 394 CALL SLAMRG( N1, N2, D, 1, -1, INDXQ ) 395 ELSE 396 QPTR( CURR+1 ) = QPTR( CURR ) 397 DO 20 I = 1, N 398 INDXQ( I ) = I 399 20 CONTINUE 400 END IF 401* 402 30 CONTINUE 403 RETURN 404* 405* End of SLAED7 406* 407 END 408