1SSTEBZ(1) LAPACK routine (version 3.1) SSTEBZ(1)
2
6 SSTEBZ - the eigenvalues of a symmetric tridiagonal matrix T
7
9 SUBROUTINE SSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M,
10 NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO )
11 CHARACTER ORDER, RANGE
13 INTEGER IL, INFO, IU, M, N, NSPLIT
15 REAL ABSTOL, VL, VU
17 INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
19 REAL D( * ), E( * ), W( * ), WORK( * )
21
23 SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T.
24 The user may ask for all eigenvalues, all eigenvalues in the half-open
25 interval (VL, VU], or the IL-th through IU-th eigenvalues.
26 To avoid overflow, the matrix must be scaled so that its
28 largest element is no greater than overflow**(1/2) *
29 underflow**(1/4) in absolute value, and for greatest
30 accuracy, it should not be much smaller than that.
31 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
33 Report CS41, Computer Science Dept., Stanford
34 University, July 21, 1966.
35
38 RANGE (input) CHARACTER*1
39 = 'A': ("All") all eigenvalues will be found.
40 = 'V': ("Value") all eigenvalues in the half-open interval (VL,
41 VU] will be found. = 'I': ("Index") the IL-th through IU-th
42 eigenvalues (of the entire matrix) will be found.
43 ORDER (input) CHARACTER*1
45 = 'B': ("By Block") the eigenvalues will be grouped by split-
46 off block (see IBLOCK, ISPLIT) and ordered from smallest to
47 largest within the block. = 'E': ("Entire matrix") the eigen‐
48 values for the entire matrix will be ordered from smallest to
49 largest.
50 N (input) INTEGER
52 The order of the tridiagonal matrix T. N >= 0.
53 VL (input) REAL
55 VU (input) REAL If RANGE='V', the lower and upper bounds
56 of the interval to be searched for eigenvalues. Eigenvalues
57 less than or equal to VL, or greater than VU, will not be
58 returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.
59 IL (input) INTEGER
61 IU (input) INTEGER If RANGE='I', the indices (in ascending
62 order) of the smallest and largest eigenvalues to be returned.
63 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
64 referenced if RANGE = 'A' or 'V'.
65 ABSTOL (input) REAL
67 The absolute tolerance for the eigenvalues. An eigenvalue (or
68 cluster) is considered to be located if it has been determined
69 to lie in an interval whose width is ABSTOL or less. If ABSTOL
70 is less than or equal to zero, then ULP*|T| will be used, where
71 |T| means the 1-norm of T.
72 Eigenvalues will be computed most accurately when ABSTOL is set
74 to twice the underflow threshold 2*SLAMCH('S'), not zero.
75 D (input) REAL array, dimension (N)
77 The n diagonal elements of the tridiagonal matrix T.
78 E (input) REAL array, dimension (N-1)
80 The (n-1) off-diagonal elements of the tridiagonal matrix T.
81 M (output) INTEGER
83 The actual number of eigenvalues found. 0 <= M <= N. (See also
84 the description of INFO=2,3.)
85 NSPLIT (output) INTEGER
87 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
88 N.
89 W (output) REAL array, dimension (N)
91 On exit, the first M elements of W will contain the eigenval‐
92 ues. (SSTEBZ may use the remaining N-M elements as workspace.)
93 IBLOCK (output) INTEGER array, dimension (N)
95 At each row/column j where E(j) is zero or small, the matrix T
96 is considered to split into a block diagonal matrix. On exit,
97 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
98 number of blocks) the eigenvalue W(i) belongs. (SSTEBZ may use
99 the remaining N-M elements as workspace.)
100 ISPLIT (output) INTEGER array, dimension (N)
102 The splitting points, at which T breaks up into submatrices.
103 The first submatrix consists of rows/columns 1 to ISPLIT(1),
104 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
105 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
106 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
107 actually be used, but since the user cannot know a priori what
108 value NSPLIT will have, N words must be reserved for ISPLIT.)
109 WORK (workspace) REAL array, dimension (4*N)
111 IWORK (workspace) INTEGER array, dimension (3*N)
113 INFO (output) INTEGER
115 = 0: successful exit
116 < 0: if INFO = -i, the i-th argument had an illegal value
117 > 0: some or all of the eigenvalues failed to converge or
118 were not computed:
119 =1 or 3: Bisection failed to converge for some eigenvalues;
120 these eigenvalues are flagged by a negative block number. The
121 effect is that the eigenvalues may not be as accurate as the
122 absolute and relative tolerances. This is generally caused by
123 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
124 Not all of the eigenvalues
125 IL:IU were found.
126 Effect: M < IU+1-IL
127 Cause: non-monotonic arithmetic, causing the Sturm sequence to
128 be non-monotonic. Cure: recalculate, using RANGE='A', and
129 pick
130 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
131 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
132 shgorin interval initially used was too small. No eigenvalues
133 were computed. Probable cause: your machine has sloppy float‐
134 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
135 recompile, and try again.
136
138 RELFAC REAL, default = 2.0e0
139 The relative tolerance. An interval (a,b] lies within "rela‐
140 tive tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp"
141 is the machine precision (distance from 1 to the next larger
142 floating point number.)
143 FUDGE REAL, default = 2
145 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
146 value of 1 should work, but on machines with sloppy arithmetic,
147 this needs to be larger. The default for publicly released
148 versions should be large enough to handle the worst machine
149 around. Note that this has no effect on accuracy of the solu‐
150 tion.
151 LAPACK routine (version 3.1) November 2006 SSTEBZ(1)