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---
:name: slarre
:md5sum: 98a7ef095c3e263ebe5088ca21b69c1a
:category: :subroutine
:arguments:
- range:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- vl:
:type: real
:intent: input/output
- vu:
:type: real
:intent: input/output
- il:
:type: integer
:intent: input
- iu:
:type: integer
:intent: input
- d:
:type: real
:intent: input/output
:dims:
- n
- e:
:type: real
:intent: input/output
:dims:
- n
- e2:
:type: real
:intent: input/output
:dims:
- n
- rtol1:
:type: real
:intent: input
- rtol2:
:type: real
:intent: input
- spltol:
:type: real
:intent: input
- nsplit:
:type: integer
:intent: output
- isplit:
:type: integer
:intent: output
:dims:
- n
- m:
:type: integer
:intent: output
- w:
:type: real
:intent: output
:dims:
- n
- werr:
:type: real
:intent: output
:dims:
- n
- wgap:
:type: real
:intent: output
:dims:
- n
- iblock:
:type: integer
:intent: output
:dims:
- n
- indexw:
:type: integer
:intent: output
:dims:
- n
- gers:
:type: real
:intent: output
:dims:
- 2*n
- pivmin:
:type: real
:intent: output
- work:
:type: real
:intent: workspace
:dims:
- 6*n
- iwork:
:type: integer
:intent: workspace
:dims:
- 5*n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* To find the desired eigenvalues of a given real symmetric\n\
* tridiagonal matrix T, SLARRE sets any \"small\" off-diagonal\n\
* elements to zero, and for each unreduced block T_i, it finds\n\
* (a) a suitable shift at one end of the block's spectrum,\n\
* (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and\n\
* (c) eigenvalues of each L_i D_i L_i^T.\n\
* The representations and eigenvalues found are then used by\n\
* SSTEMR to compute the eigenvectors of T.\n\
* The accuracy varies depending on whether bisection is used to\n\
* find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to\n\
* conpute all and then discard any unwanted one.\n\
* As an added benefit, SLARRE also outputs the n\n\
* Gerschgorin intervals for the matrices L_i D_i L_i^T.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* RANGE (input) CHARACTER*1\n\
* = 'A': (\"All\") all eigenvalues will be found.\n\
* = 'V': (\"Value\") all eigenvalues in the half-open interval\n\
* (VL, VU] will be found.\n\
* = 'I': (\"Index\") the IL-th through IU-th eigenvalues (of the\n\
* entire matrix) will be found.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix. N > 0.\n\
*\n\
* VL (input/output) REAL \n\
* VU (input/output) REAL \n\
* If RANGE='V', the lower and upper bounds for the eigenvalues.\n\
* Eigenvalues less than or equal to VL, or greater than VU,\n\
* will not be returned. VL < VU.\n\
* If RANGE='I' or ='A', SLARRE computes bounds on the desired\n\
* part of the spectrum.\n\
*\n\
* IL (input) INTEGER\n\
* IU (input) INTEGER\n\
* If RANGE='I', the indices (in ascending order) of the\n\
* smallest and largest eigenvalues to be returned.\n\
* 1 <= IL <= IU <= N.\n\
*\n\
* D (input/output) REAL array, dimension (N)\n\
* On entry, the N diagonal elements of the tridiagonal\n\
* matrix T.\n\
* On exit, the N diagonal elements of the diagonal\n\
* matrices D_i.\n\
*\n\
* E (input/output) REAL array, dimension (N)\n\
* On entry, the first (N-1) entries contain the subdiagonal\n\
* elements of the tridiagonal matrix T; E(N) need not be set.\n\
* On exit, E contains the subdiagonal elements of the unit\n\
* bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),\n\
* 1 <= I <= NSPLIT, contain the base points sigma_i on output.\n\
*\n\
* E2 (input/output) REAL array, dimension (N)\n\
* On entry, the first (N-1) entries contain the SQUARES of the\n\
* subdiagonal elements of the tridiagonal matrix T;\n\
* E2(N) need not be set.\n\
* On exit, the entries E2( ISPLIT( I ) ),\n\
* 1 <= I <= NSPLIT, have been set to zero\n\
*\n\
* RTOL1 (input) REAL \n\
* RTOL2 (input) REAL \n\
* Parameters for bisection.\n\
* An interval [LEFT,RIGHT] has converged if\n\
* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )\n\
*\n\
* SPLTOL (input) REAL \n\
* The threshold for splitting.\n\
*\n\
* NSPLIT (output) INTEGER\n\
* The number of blocks T splits into. 1 <= NSPLIT <= N.\n\
*\n\
* ISPLIT (output) INTEGER array, dimension (N)\n\
* The splitting points, at which T breaks up into blocks.\n\
* The first block consists of rows/columns 1 to ISPLIT(1),\n\
* the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),\n\
* etc., and the NSPLIT-th consists of rows/columns\n\
* ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.\n\
*\n\
* M (output) INTEGER\n\
* The total number of eigenvalues (of all L_i D_i L_i^T)\n\
* found.\n\
*\n\
* W (output) REAL array, dimension (N)\n\
* The first M elements contain the eigenvalues. The\n\
* eigenvalues of each of the blocks, L_i D_i L_i^T, are\n\
* sorted in ascending order ( SLARRE may use the\n\
* remaining N-M elements as workspace).\n\
*\n\
* WERR (output) REAL array, dimension (N)\n\
* The error bound on the corresponding eigenvalue in W.\n\
*\n\
* WGAP (output) REAL array, dimension (N)\n\
* The separation from the right neighbor eigenvalue in W.\n\
* The gap is only with respect to the eigenvalues of the same block\n\
* as each block has its own representation tree.\n\
* Exception: at the right end of a block we store the left gap\n\
*\n\
* IBLOCK (output) INTEGER array, dimension (N)\n\
* The indices of the blocks (submatrices) associated with the\n\
* corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue\n\
* W(i) belongs to the first block from the top, =2 if W(i)\n\
* belongs to the second block, etc.\n\
*\n\
* INDEXW (output) INTEGER array, dimension (N)\n\
* The indices of the eigenvalues within each block (submatrix);\n\
* for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the\n\
* i-th eigenvalue W(i) is the 10-th eigenvalue in block 2\n\
*\n\
* GERS (output) REAL array, dimension (2*N)\n\
* The N Gerschgorin intervals (the i-th Gerschgorin interval\n\
* is (GERS(2*i-1), GERS(2*i)).\n\
*\n\
* PIVMIN (output) REAL\n\
* The minimum pivot in the Sturm sequence for T.\n\
*\n\
* WORK (workspace) REAL array, dimension (6*N)\n\
* Workspace.\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (5*N)\n\
* Workspace.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* > 0: A problem occurred in SLARRE.\n\
* < 0: One of the called subroutines signaled an internal problem.\n\
* Needs inspection of the corresponding parameter IINFO\n\
* for further information.\n\
*\n\
* =-1: Problem in SLARRD.\n\
* = 2: No base representation could be found in MAXTRY iterations.\n\
* Increasing MAXTRY and recompilation might be a remedy.\n\
* =-3: Problem in SLARRB when computing the refined root\n\
* representation for SLASQ2.\n\
* =-4: Problem in SLARRB when preforming bisection on the\n\
* desired part of the spectrum.\n\
* =-5: Problem in SLASQ2.\n\
* =-6: Problem in SLASQ2.\n\
*\n\n\
* Further Details\n\
* The base representations are required to suffer very little\n\
* element growth and consequently define all their eigenvalues to\n\
* high relative accuracy.\n\
* ===============\n\
*\n\
* Based on contributions by\n\
* Beresford Parlett, University of California, Berkeley, USA\n\
* Jim Demmel, University of California, Berkeley, USA\n\
* Inderjit Dhillon, University of Texas, Austin, USA\n\
* Osni Marques, LBNL/NERSC, USA\n\
* Christof Voemel, University of California, Berkeley, USA\n\
*\n\
* =====================================================================\n\
*\n"
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