1SLARRD(1) LAPACK auxiliary routine (version 3.2) SLARRD(1)
2
6 SLARRD - computes the eigenvalues of a symmetric tridiagonal matrix T
7 to suitable 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
32 largest element is no greater than overflow**(1/2) *
33 underflow**(1/4) in absolute value, and for greatest
34 accuracy, it should not be much smaller than that.
35 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
36 Report CS41, Computer Science Dept., Stanford
37 University, July 21, 1966.
38
40 RANGE (input) CHARACTER
41 = 'A': ("All") all eigenvalues will be found.
42 = 'V': ("Value") all eigenvalues in the half-open interval (VL,
43 VU] will be found. = 'I': ("Index") the IL-th through IU-th
44 eigenvalues (of the entire matrix) will be found.
45 ORDER (input) CHARACTER
47 = 'B': ("By Block") the eigenvalues will be grouped by split-
48 off block (see IBLOCK, ISPLIT) and ordered from smallest to
49 largest within the block. = 'E': ("Entire matrix") the eigen‐
50 values for the entire matrix will be ordered from smallest to
51 largest.
52 N (input) INTEGER
54 The order of the tridiagonal matrix T. N >= 0.
55 VL (input) REAL
57 VU (input) REAL If RANGE='V', the lower and upper bounds
58 of the interval to be searched for eigenvalues. Eigenvalues
59 less than or equal to VL, or greater than VU, will not be
60 returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.
61 IL (input) INTEGER
63 IU (input) INTEGER If RANGE='I', the indices (in ascending
64 order) of the smallest and largest eigenvalues to be returned.
65 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
66 referenced if RANGE = 'A' or 'V'.
67 GERS (input) REAL array, dimension (2*N)
69 The N Gerschgorin intervals (the i-th Gerschgorin interval is
70 (GERS(2*i-1), GERS(2*i)).
71 RELTOL (input) REAL
73 The minimum relative width of an interval. When an interval is
74 narrower than RELTOL times the larger (in magnitude) endpoint,
75 then it is considered to be sufficiently small, i.e., con‐
76 verged. Note: this should always be at least radix*machine
77 epsilon.
78 D (input) REAL array, dimension (N)
80 The n diagonal elements of the tridiagonal matrix T.
81 E (input) REAL array, dimension (N-1)
83 The (n-1) off-diagonal elements of the tridiagonal matrix T.
84 E2 (input) REAL array, dimension (N-1)
86 The (n-1) squared off-diagonal elements of the tridiagonal
87 matrix T.
88 PIVMIN (input) REAL
90 The minimum pivot allowed in the Sturm sequence for T.
91 NSPLIT (input) INTEGER
93 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
94 N.
95 ISPLIT (input) INTEGER array, dimension (N)
97 The splitting points, at which T breaks up into submatrices.
98 The first submatrix consists of rows/columns 1 to ISPLIT(1),
99 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
100 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
101 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
102 actually be used, but since the user cannot know a priori what
103 value NSPLIT will have, N words must be reserved for ISPLIT.)
104 M (output) INTEGER
106 The actual number of eigenvalues found. 0 <= M <= N. (See also
107 the description of INFO=2,3.)
108 W (output) REAL array, dimension (N)
110 On exit, the first M elements of W will contain the eigenvalue
111 approximations. SLARRD computes an interval I_j = (a_j, b_j]
112 that includes eigenvalue j. The eigenvalue approximation is
113 given as the interval midpoint W(j)= ( a_j + b_j)/2. The corre‐
114 sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
115 WERR (output) REAL array, dimension (N)
117 The error bound on the corresponding eigenvalue approximation
118 in W.
119 WL (output) REAL
121 WU (output) REAL The interval (WL, WU] contains all the
122 wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If
123 RANGE='A', then WL and WU are the global Gerschgorin bounds on
124 the spectrum. If RANGE='I', then WL and WU are computed by
125 SLAEBZ from the index range specified.
126 IBLOCK (output) INTEGER array, dimension (N)
128 At each row/column j where E(j) is zero or small, the matrix T
129 is considered to split into a block diagonal matrix. On exit,
130 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
131 number of blocks) the eigenvalue W(i) belongs. (SLARRD may use
132 the remaining N-M elements as workspace.)
133 INDEXW (output) INTEGER array, dimension (N)
135 The indices of the eigenvalues within each block (submatrix);
136 for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th
137 eigenvalue W(i) is the j-th eigenvalue in block k.
138 WORK (workspace) REAL array, dimension (4*N)
140 IWORK (workspace) INTEGER array, dimension (3*N)
142 INFO (output) INTEGER
144 = 0: successful exit
145 < 0: if INFO = -i, the i-th argument had an illegal value
146 > 0: some or all of the eigenvalues failed to converge or
147 were not computed:
148 =1 or 3: Bisection failed to converge for some eigenvalues;
149 these eigenvalues are flagged by a negative block number. The
150 effect is that the eigenvalues may not be as accurate as the
151 absolute and relative tolerances. This is generally caused by
152 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
153 Not all of the eigenvalues
154 IL:IU were found.
155 Effect: M < IU+1-IL
156 Cause: non-monotonic arithmetic, causing the Sturm sequence to
157 be non-monotonic. Cure: recalculate, using RANGE='A', and
158 pick
159 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
160 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
161 shgorin interval initially used was too small. No eigenvalues
162 were computed. Probable cause: your machine has sloppy float‐
163 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
164 recompile, and try again.
165
167 FUDGE REAL , default = 2
168 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
169 value of 1 should work, but on machines with sloppy arithmetic,
170 this needs to be larger. The default for publicly released
171 versions should be large enough to handle the worst machine
172 around. Note that this has no effect on accuracy of the solu‐
173 tion. Based on contributions by W. Kahan, University of Cali‐
174 fornia, Berkeley, USA Beresford Parlett, University of Califor‐
175 nia, Berkeley, USA Jim Demmel, University of California, Berke‐
176 ley, USA Inderjit Dhillon, University of Texas, Austin, USA
177 Osni Marques, LBNL/NERSC, USA Christof Voemel, University of
178 California, Berkeley, USA
179 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLARRD(1)