1SLARRD(1) LAPACK auxiliary routine (version 3.1) SLARRD(1)
2
6 SLARRD - the eigenvalues of a symmetric tridiagonal matrix T to suit‐
7 able accuracy
8
10 SUBROUTINE SLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E,
11 E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU,
12 IBLOCK, INDEXW, WORK, IWORK, INFO )
13 CHARACTER ORDER, RANGE
15 INTEGER IL, INFO, IU, M, N, NSPLIT
17 REAL PIVMIN, RELTOL, VL, VU, WL, WU
19 INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), IWORK( * )
21 REAL D( * ), E( * ), E2( * ), GERS( * ), W( * ), WERR( *
23 ), WORK( * )
24
26 SLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to
27 suitable accuracy. This is an auxiliary code to be called from SSTEMR.
28 The user may ask for all eigenvalues, all eigenvalues
29 in the half-open interval (VL, VU], or the IL-th through IU-th eigen‐
30 values.
31 To avoid overflow, the matrix must be scaled so that its
33 largest element is no greater than overflow**(1/2) *
34 underflow**(1/4) in absolute value, and for greatest
35 accuracy, it should not be much smaller than that.
36 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
38 Report CS41, Computer Science Dept., Stanford
39 University, July 21, 1966.
40
43 RANGE (input) CHARACTER
44 = 'A': ("All") all eigenvalues will be found.
45 = 'V': ("Value") all eigenvalues in the half-open interval (VL,
46 VU] will be found. = 'I': ("Index") the IL-th through IU-th
47 eigenvalues (of the entire matrix) will be found.
48 ORDER (input) CHARACTER
50 = 'B': ("By Block") the eigenvalues will be grouped by split-
51 off block (see IBLOCK, ISPLIT) and ordered from smallest to
52 largest within the block. = 'E': ("Entire matrix") the eigen‐
53 values for the entire matrix will be ordered from smallest to
54 largest.
55 N (input) INTEGER
57 The order of the tridiagonal matrix T. N >= 0.
58 VL (input) REAL
60 VU (input) REAL If RANGE='V', the lower and upper bounds
61 of the interval to be searched for eigenvalues. Eigenvalues
62 less than or equal to VL, or greater than VU, will not be
63 returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.
64 IL (input) INTEGER
66 IU (input) INTEGER If RANGE='I', the indices (in ascending
67 order) of the smallest and largest eigenvalues to be returned.
68 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
69 referenced if RANGE = 'A' or 'V'.
70 GERS (input) REAL array, dimension (2*N)
72 The N Gerschgorin intervals (the i-th Gerschgorin interval is
73 (GERS(2*i-1), GERS(2*i)).
74 RELTOL (input) REAL
76 The minimum relative width of an interval. When an interval is
77 narrower than RELTOL times the larger (in magnitude) endpoint,
78 then it is considered to be sufficiently small, i.e., con‐
79 verged. Note: this should always be at least radix*machine
80 epsilon.
81 D (input) REAL array, dimension (N)
83 The n diagonal elements of the tridiagonal matrix T.
84 E (input) REAL array, dimension (N-1)
86 The (n-1) off-diagonal elements of the tridiagonal matrix T.
87 E2 (input) REAL array, dimension (N-1)
89 The (n-1) squared off-diagonal elements of the tridiagonal
90 matrix T.
91 PIVMIN (input) REAL
93 The minimum pivot allowed in the Sturm sequence for T.
94 NSPLIT (input) INTEGER
96 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
97 N.
98 ISPLIT (input) INTEGER array, dimension (N)
100 The splitting points, at which T breaks up into submatrices.
101 The first submatrix consists of rows/columns 1 to ISPLIT(1),
102 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
103 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
104 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
105 actually be used, but since the user cannot know a priori what
106 value NSPLIT will have, N words must be reserved for ISPLIT.)
107 M (output) INTEGER
109 The actual number of eigenvalues found. 0 <= M <= N. (See also
110 the description of INFO=2,3.)
111 W (output) REAL array, dimension (N)
113 On exit, the first M elements of W will contain the eigenvalue
114 approximations. SLARRD computes an interval I_j = (a_j, b_j]
115 that includes eigenvalue j. The eigenvalue approximation is
116 given as the interval midpoint W(j)= ( a_j + b_j)/2. The corre‐
117 sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
118 WERR (output) REAL array, dimension (N)
120 The error bound on the corresponding eigenvalue approximation
121 in W.
122 WL (output) REAL
124 WU (output) REAL The interval (WL, WU] contains all the
125 wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If
126 RANGE='A', then WL and WU are the global Gerschgorin bounds on
127 the spectrum. If RANGE='I', then WL and WU are computed by
128 SLAEBZ from the index range specified.
129 IBLOCK (output) INTEGER array, dimension (N)
131 At each row/column j where E(j) is zero or small, the matrix T
132 is considered to split into a block diagonal matrix. On exit,
133 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
134 number of blocks) the eigenvalue W(i) belongs. (SLARRD may use
135 the remaining N-M elements as workspace.)
136 INDEXW (output) INTEGER array, dimension (N)
138 The indices of the eigenvalues within each block (submatrix);
139 for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th
140 eigenvalue W(i) is the j-th eigenvalue in block k.
141 WORK (workspace) REAL array, dimension (4*N)
143 IWORK (workspace) INTEGER array, dimension (3*N)
145 INFO (output) INTEGER
147 = 0: successful exit
148 < 0: if INFO = -i, the i-th argument had an illegal value
149 > 0: some or all of the eigenvalues failed to converge or
150 were not computed:
151 =1 or 3: Bisection failed to converge for some eigenvalues;
152 these eigenvalues are flagged by a negative block number. The
153 effect is that the eigenvalues may not be as accurate as the
154 absolute and relative tolerances. This is generally caused by
155 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
156 Not all of the eigenvalues
157 IL:IU were found.
158 Effect: M < IU+1-IL
159 Cause: non-monotonic arithmetic, causing the Sturm sequence to
160 be non-monotonic. Cure: recalculate, using RANGE='A', and
161 pick
162 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
163 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
164 shgorin interval initially used was too small. No eigenvalues
165 were computed. Probable cause: your machine has sloppy float‐
166 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
167 recompile, and try again.
168
170 FUDGE REAL , default = 2
171 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
172 value of 1 should work, but on machines with sloppy arithmetic,
173 this needs to be larger. The default for publicly released
174 versions should be large enough to handle the worst machine
175 around. Note that this has no effect on accuracy of the solu‐
176 tion.
177 Based on contributions by W. Kahan, University of California,
179 Berkeley, USA Beresford Parlett, University of California,
180 Berkeley, USA Jim Demmel, University of California, Berkeley,
181 USA Inderjit Dhillon, University of Texas, Austin, USA Osni
182 Marques, LBNL/NERSC, USA Christof Voemel, University of Cali‐
183 fornia, Berkeley, USA
184 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLARRD(1)