1DLARRD(1) LAPACK auxiliary routine (version 3.2) DLARRD(1)
2
6 DLARRD - computes the eigenvalues of a symmetric tridiagonal matrix T
7 to suitable accuracy
8
10 SUBROUTINE DLARRD( 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 DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU
19 INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), IWORK( * )
21 DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ), W( *
23 ), WERR( * ), WORK( * )
24
26 DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to
27 suitable accuracy. This is an auxiliary code to be called from DSTEMR.
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) DOUBLE PRECISION
57 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
58 upper bounds of the interval to be searched for eigenvalues.
59 Eigenvalues less than or equal to VL, or greater than VU, will
60 not be returned. VL < VU. Not referenced if RANGE = 'A' or
61 'I'.
62 IL (input) INTEGER
64 IU (input) INTEGER If RANGE='I', the indices (in ascending
65 order) of the smallest and largest eigenvalues to be returned.
66 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
67 referenced if RANGE = 'A' or 'V'.
68 GERS (input) DOUBLE PRECISION array, dimension (2*N)
70 The N Gerschgorin intervals (the i-th Gerschgorin interval is
71 (GERS(2*i-1), GERS(2*i)).
72 RELTOL (input) DOUBLE PRECISION
74 The minimum relative width of an interval. When an interval is
75 narrower than RELTOL times the larger (in magnitude) endpoint,
76 then it is considered to be sufficiently small, i.e., con‐
77 verged. Note: this should always be at least radix*machine
78 epsilon.
79 D (input) DOUBLE PRECISION array, dimension (N)
81 The n diagonal elements of the tridiagonal matrix T.
82 E (input) DOUBLE PRECISION array, dimension (N-1)
84 The (n-1) off-diagonal elements of the tridiagonal matrix T.
85 E2 (input) DOUBLE PRECISION array, dimension (N-1)
87 The (n-1) squared off-diagonal elements of the tridiagonal
88 matrix T.
89 PIVMIN (input) DOUBLE PRECISION
91 The minimum pivot allowed in the Sturm sequence for T.
92 NSPLIT (input) INTEGER
94 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
95 N.
96 ISPLIT (input) INTEGER array, dimension (N)
98 The splitting points, at which T breaks up into submatrices.
99 The first submatrix consists of rows/columns 1 to ISPLIT(1),
100 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
101 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
102 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
103 actually be used, but since the user cannot know a priori what
104 value NSPLIT will have, N words must be reserved for ISPLIT.)
105 M (output) INTEGER
107 The actual number of eigenvalues found. 0 <= M <= N. (See also
108 the description of INFO=2,3.)
109 W (output) DOUBLE PRECISION array, dimension (N)
111 On exit, the first M elements of W will contain the eigenvalue
112 approximations. DLARRD computes an interval I_j = (a_j, b_j]
113 that includes eigenvalue j. The eigenvalue approximation is
114 given as the interval midpoint W(j)= ( a_j + b_j)/2. The corre‐
115 sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
116 WERR (output) DOUBLE PRECISION array, dimension (N)
118 The error bound on the corresponding eigenvalue approximation
119 in W.
120 WL (output) DOUBLE PRECISION
122 WU (output) DOUBLE PRECISION The interval (WL, WU] con‐
123 tains all the wanted eigenvalues. If RANGE='V', then WL=VL and
124 WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin
125 bounds on the spectrum. If RANGE='I', then WL and WU are com‐
126 puted by DLAEBZ from the index range specified.
127 IBLOCK (output) INTEGER array, dimension (N)
129 At each row/column j where E(j) is zero or small, the matrix T
130 is considered to split into a block diagonal matrix. On exit,
131 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
132 number of blocks) the eigenvalue W(i) belongs. (DLARRD may use
133 the remaining N-M elements as workspace.)
134 INDEXW (output) INTEGER array, dimension (N)
136 The indices of the eigenvalues within each block (submatrix);
137 for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th
138 eigenvalue W(i) is the j-th eigenvalue in block k.
139 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
141 IWORK (workspace) INTEGER array, dimension (3*N)
143 INFO (output) INTEGER
145 = 0: successful exit
146 < 0: if INFO = -i, the i-th argument had an illegal value
147 > 0: some or all of the eigenvalues failed to converge or
148 were not computed:
149 =1 or 3: Bisection failed to converge for some eigenvalues;
150 these eigenvalues are flagged by a negative block number. The
151 effect is that the eigenvalues may not be as accurate as the
152 absolute and relative tolerances. This is generally caused by
153 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
154 Not all of the eigenvalues
155 IL:IU were found.
156 Effect: M < IU+1-IL
157 Cause: non-monotonic arithmetic, causing the Sturm sequence to
158 be non-monotonic. Cure: recalculate, using RANGE='A', and
159 pick
160 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
161 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
162 shgorin interval initially used was too small. No eigenvalues
163 were computed. Probable cause: your machine has sloppy float‐
164 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
165 recompile, and try again.
166
168 FUDGE DOUBLE PRECISION, default = 2
169 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
170 value of 1 should work, but on machines with sloppy arithmetic,
171 this needs to be larger. The default for publicly released
172 versions should be large enough to handle the worst machine
173 around. Note that this has no effect on accuracy of the solu‐
174 tion. Based on contributions by W. Kahan, University of Cali‐
175 fornia, Berkeley, USA Beresford Parlett, University of Califor‐
176 nia, Berkeley, USA Jim Demmel, University of California, Berke‐
177 ley, USA Inderjit Dhillon, University of Texas, Austin, USA
178 Osni Marques, LBNL/NERSC, USA Christof Voemel, University of
179 California, Berkeley, USA
180 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLARRD(1)