1DLARRD(1) LAPACK auxiliary routine (version 3.1) DLARRD(1)
2
6 DLARRD - the eigenvalues of a symmetric tridiagonal matrix T to suit‐
7 able 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
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) DOUBLE PRECISION
60 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
61 upper bounds of the interval to be searched for eigenvalues.
62 Eigenvalues less than or equal to VL, or greater than VU, will
63 not be returned. VL < VU. Not referenced if RANGE = 'A' or
64 'I'.
65 IL (input) INTEGER
67 IU (input) INTEGER If RANGE='I', the indices (in ascending
68 order) of the smallest and largest eigenvalues to be returned.
69 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
70 referenced if RANGE = 'A' or 'V'.
71 GERS (input) DOUBLE PRECISION array, dimension (2*N)
73 The N Gerschgorin intervals (the i-th Gerschgorin interval is
74 (GERS(2*i-1), GERS(2*i)).
75 RELTOL (input) DOUBLE PRECISION
77 The minimum relative width of an interval. When an interval is
78 narrower than RELTOL times the larger (in magnitude) endpoint,
79 then it is considered to be sufficiently small, i.e., con‐
80 verged. Note: this should always be at least radix*machine
81 epsilon.
82 D (input) DOUBLE PRECISION array, dimension (N)
84 The n diagonal elements of the tridiagonal matrix T.
85 E (input) DOUBLE PRECISION array, dimension (N-1)
87 The (n-1) off-diagonal elements of the tridiagonal matrix T.
88 E2 (input) DOUBLE PRECISION array, dimension (N-1)
90 The (n-1) squared off-diagonal elements of the tridiagonal
91 matrix T.
92 PIVMIN (input) DOUBLE PRECISION
94 The minimum pivot allowed in the Sturm sequence for T.
95 NSPLIT (input) INTEGER
97 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
98 N.
99 ISPLIT (input) INTEGER array, dimension (N)
101 The splitting points, at which T breaks up into submatrices.
102 The first submatrix consists of rows/columns 1 to ISPLIT(1),
103 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
104 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
105 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
106 actually be used, but since the user cannot know a priori what
107 value NSPLIT will have, N words must be reserved for ISPLIT.)
108 M (output) INTEGER
110 The actual number of eigenvalues found. 0 <= M <= N. (See also
111 the description of INFO=2,3.)
112 W (output) DOUBLE PRECISION array, dimension (N)
114 On exit, the first M elements of W will contain the eigenvalue
115 approximations. DLARRD computes an interval I_j = (a_j, b_j]
116 that includes eigenvalue j. The eigenvalue approximation is
117 given as the interval midpoint W(j)= ( a_j + b_j)/2. The corre‐
118 sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
119 WERR (output) DOUBLE PRECISION array, dimension (N)
121 The error bound on the corresponding eigenvalue approximation
122 in W.
123 WL (output) DOUBLE PRECISION
125 WU (output) DOUBLE PRECISION The interval (WL, WU] con‐
126 tains all the wanted eigenvalues. If RANGE='V', then WL=VL and
127 WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin
128 bounds on the spectrum. If RANGE='I', then WL and WU are com‐
129 puted by DLAEBZ from the index range specified.
130 IBLOCK (output) INTEGER array, dimension (N)
132 At each row/column j where E(j) is zero or small, the matrix T
133 is considered to split into a block diagonal matrix. On exit,
134 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
135 number of blocks) the eigenvalue W(i) belongs. (DLARRD may use
136 the remaining N-M elements as workspace.)
137 INDEXW (output) INTEGER array, dimension (N)
139 The indices of the eigenvalues within each block (submatrix);
140 for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th
141 eigenvalue W(i) is the j-th eigenvalue in block k.
142 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
144 IWORK (workspace) INTEGER array, dimension (3*N)
146 INFO (output) INTEGER
148 = 0: successful exit
149 < 0: if INFO = -i, the i-th argument had an illegal value
150 > 0: some or all of the eigenvalues failed to converge or
151 were not computed:
152 =1 or 3: Bisection failed to converge for some eigenvalues;
153 these eigenvalues are flagged by a negative block number. The
154 effect is that the eigenvalues may not be as accurate as the
155 absolute and relative tolerances. This is generally caused by
156 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
157 Not all of the eigenvalues
158 IL:IU were found.
159 Effect: M < IU+1-IL
160 Cause: non-monotonic arithmetic, causing the Sturm sequence to
161 be non-monotonic. Cure: recalculate, using RANGE='A', and
162 pick
163 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
164 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
165 shgorin interval initially used was too small. No eigenvalues
166 were computed. Probable cause: your machine has sloppy float‐
167 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
168 recompile, and try again.
169
171 FUDGE DOUBLE PRECISION, default = 2
172 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
173 value of 1 should work, but on machines with sloppy arithmetic,
174 this needs to be larger. The default for publicly released
175 versions should be large enough to handle the worst machine
176 around. Note that this has no effect on accuracy of the solu‐
177 tion.
178 Based on contributions by W. Kahan, University of California,
180 Berkeley, USA Beresford Parlett, University of California,
181 Berkeley, USA Jim Demmel, University of California, Berkeley,
182 USA Inderjit Dhillon, University of Texas, Austin, USA Osni
183 Marques, LBNL/NERSC, USA Christof Voemel, University of Cali‐
184 fornia, Berkeley, USA
185 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 DLARRD(1)