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---
:name: slarrd
:md5sum: 1d17cb93ea8bc4bff6584301e13e6ec7
:category: :subroutine
:arguments:
- range:
:type: char
:intent: input
- order:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- vl:
:type: real
:intent: input
- vu:
:type: real
:intent: input
- il:
:type: integer
:intent: input
- iu:
:type: integer
:intent: input
- gers:
:type: real
:intent: input
:dims:
- 2*n
- reltol:
:type: real
:intent: input
- d:
:type: real
:intent: input
:dims:
- n
- e:
:type: real
:intent: input
:dims:
- n-1
- e2:
:type: real
:intent: input
:dims:
- n-1
- pivmin:
:type: real
:intent: input
- nsplit:
:type: integer
:intent: input
- isplit:
:type: integer
:intent: input
:dims:
- n
- m:
:type: integer
:intent: output
- w:
:type: real
:intent: output
:dims:
- n
- werr:
:type: real
:intent: output
:dims:
- n
- wl:
:type: real
:intent: output
- wu:
:type: real
:intent: output
- iblock:
:type: integer
:intent: output
:dims:
- n
- indexw:
:type: integer
:intent: output
:dims:
- n
- work:
:type: real
:intent: workspace
:dims:
- 4*n
- iwork:
:type: integer
:intent: workspace
:dims:
- 3*n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE SLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* SLARRD computes the eigenvalues of a symmetric tridiagonal\n\
* matrix T to suitable accuracy. This is an auxiliary code to be\n\
* called from SSTEMR.\n\
* The user may ask for all eigenvalues, all eigenvalues\n\
* in the half-open interval (VL, VU], or the IL-th through IU-th\n\
* eigenvalues.\n\
*\n\
* To avoid overflow, the matrix must be scaled so that its\n\
* largest element is no greater than overflow**(1/2) *\n\
* underflow**(1/4) in absolute value, and for greatest\n\
* accuracy, it should not be much smaller than that.\n\
*\n\
* See W. Kahan \"Accurate Eigenvalues of a Symmetric Tridiagonal\n\
* Matrix\", Report CS41, Computer Science Dept., Stanford\n\
* University, July 21, 1966.\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\
* ORDER (input) CHARACTER*1\n\
* = 'B': (\"By Block\") the eigenvalues will be grouped by\n\
* split-off block (see IBLOCK, ISPLIT) and\n\
* ordered from smallest to largest within\n\
* the block.\n\
* = 'E': (\"Entire matrix\")\n\
* the eigenvalues for the entire matrix\n\
* will be ordered from smallest to\n\
* largest.\n\
*\n\
* N (input) INTEGER\n\
* The order of the tridiagonal matrix T. N >= 0.\n\
*\n\
* VL (input) REAL \n\
* VU (input) REAL \n\
* If RANGE='V', the lower and upper bounds of the interval to\n\
* be searched for eigenvalues. Eigenvalues less than or equal\n\
* to VL, or greater than VU, will not be returned. VL < VU.\n\
* Not referenced if RANGE = 'A' or 'I'.\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, if N > 0; IL = 1 and IU = 0 if N = 0.\n\
* Not referenced if RANGE = 'A' or 'V'.\n\
*\n\
* GERS (input) 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\
* RELTOL (input) REAL \n\
* The minimum relative width of an interval. When an interval\n\
* is narrower than RELTOL times the larger (in\n\
* magnitude) endpoint, then it is considered to be\n\
* sufficiently small, i.e., converged. Note: this should\n\
* always be at least radix*machine epsilon.\n\
*\n\
* D (input) REAL array, dimension (N)\n\
* The n diagonal elements of the tridiagonal matrix T.\n\
*\n\
* E (input) REAL array, dimension (N-1)\n\
* The (n-1) off-diagonal elements of the tridiagonal matrix T.\n\
*\n\
* E2 (input) REAL array, dimension (N-1)\n\
* The (n-1) squared off-diagonal elements of the tridiagonal matrix T.\n\
*\n\
* PIVMIN (input) REAL \n\
* The minimum pivot allowed in the Sturm sequence for T.\n\
*\n\
* NSPLIT (input) INTEGER\n\
* The number of diagonal blocks in the matrix T.\n\
* 1 <= NSPLIT <= N.\n\
*\n\
* ISPLIT (input) INTEGER array, dimension (N)\n\
* The splitting points, at which T breaks up into submatrices.\n\
* The first submatrix 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\
* (Only the first NSPLIT elements will actually be used, but\n\
* since the user cannot know a priori what value NSPLIT will\n\
* have, N words must be reserved for ISPLIT.)\n\
*\n\
* M (output) INTEGER\n\
* The actual number of eigenvalues found. 0 <= M <= N.\n\
* (See also the description of INFO=2,3.)\n\
*\n\
* W (output) REAL array, dimension (N)\n\
* On exit, the first M elements of W will contain the\n\
* eigenvalue approximations. SLARRD computes an interval\n\
* I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue\n\
* approximation is given as the interval midpoint\n\
* W(j)= ( a_j + b_j)/2. The corresponding error is bounded by\n\
* WERR(j) = abs( a_j - b_j)/2\n\
*\n\
* WERR (output) REAL array, dimension (N)\n\
* The error bound on the corresponding eigenvalue approximation\n\
* in W.\n\
*\n\
* WL (output) REAL \n\
* WU (output) REAL \n\
* The interval (WL, WU] contains all the wanted eigenvalues.\n\
* If RANGE='V', then WL=VL and WU=VU.\n\
* If RANGE='A', then WL and WU are the global Gerschgorin bounds\n\
* on the spectrum.\n\
* If RANGE='I', then WL and WU are computed by SLAEBZ from the\n\
* index range specified.\n\
*\n\
* IBLOCK (output) INTEGER array, dimension (N)\n\
* At each row/column j where E(j) is zero or small, the\n\
* matrix T is considered to split into a block diagonal\n\
* matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which\n\
* block (from 1 to the number of blocks) the eigenvalue W(i)\n\
* belongs. (SLARRD may use the remaining N-M elements as\n\
* workspace.)\n\
*\n\
* INDEXW (output) INTEGER array, dimension (N)\n\
* The indices of the eigenvalues within each block (submatrix);\n\
* for example, INDEXW(i)= j and IBLOCK(i)=k imply that the\n\
* i-th eigenvalue W(i) is the j-th eigenvalue in block k.\n\
*\n\
* WORK (workspace) REAL array, dimension (4*N)\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (3*N)\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
* > 0: some or all of the eigenvalues failed to converge or\n\
* were not computed:\n\
* =1 or 3: Bisection failed to converge for some\n\
* eigenvalues; these eigenvalues are flagged by a\n\
* negative block number. The effect is that the\n\
* eigenvalues may not be as accurate as the\n\
* absolute and relative tolerances. This is\n\
* generally caused by unexpectedly inaccurate\n\
* arithmetic.\n\
* =2 or 3: RANGE='I' only: Not all of the eigenvalues\n\
* IL:IU were found.\n\
* Effect: M < IU+1-IL\n\
* Cause: non-monotonic arithmetic, causing the\n\
* Sturm sequence to be non-monotonic.\n\
* Cure: recalculate, using RANGE='A', and pick\n\
* out eigenvalues IL:IU. In some cases,\n\
* increasing the PARAMETER \"FUDGE\" may\n\
* make things work.\n\
* = 4: RANGE='I', and the Gershgorin interval\n\
* initially used was too small. No eigenvalues\n\
* were computed.\n\
* Probable cause: your machine has sloppy\n\
* floating-point arithmetic.\n\
* Cure: Increase the PARAMETER \"FUDGE\",\n\
* recompile, and try again.\n\
*\n\
* Internal Parameters\n\
* ===================\n\
*\n\
* FUDGE REAL , default = 2\n\
* A \"fudge factor\" to widen the Gershgorin intervals. Ideally,\n\
* a value of 1 should work, but on machines with sloppy\n\
* arithmetic, this needs to be larger. The default for\n\
* publicly released versions should be large enough to handle\n\
* the worst machine around. Note that this has no effect\n\
* on accuracy of the solution.\n\
*\n\
* Based on contributions by\n\
* W. Kahan, University of California, Berkeley, USA\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\
*\n"
|
