1SSTEBZ(1) LAPACK routine (version 3.2) SSTEBZ(1)
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6 SSTEBZ - computes the eigenvalues of a symmetric tridiagonal matrix T
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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( * )
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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
27 largest element is no greater than overflow**(1/2) *
28 underflow**(1/4) in absolute value, and for greatest
29 accuracy, it should not be much smaller than that.
30 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
31 Report CS41, Computer Science Dept., Stanford
32 University, July 21, 1966.
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35 RANGE (input) CHARACTER*1
36 = 'A': ("All") all eigenvalues will be found.
37 = 'V': ("Value") all eigenvalues in the half-open interval (VL,
38 VU] will be found. = 'I': ("Index") the IL-th through IU-th
39 eigenvalues (of the entire matrix) will be found.
40 ORDER (input) CHARACTER*1
42 = 'B': ("By Block") the eigenvalues will be grouped by split-
43 off block (see IBLOCK, ISPLIT) and ordered from smallest to
44 largest within the block. = 'E': ("Entire matrix") the eigen‐
45 values for the entire matrix will be ordered from smallest to
46 largest.
47 N (input) INTEGER
49 The order of the tridiagonal matrix T. N >= 0.
50 VL (input) REAL
52 VU (input) REAL If RANGE='V', the lower and upper bounds
53 of the interval to be searched for eigenvalues. Eigenvalues
54 less than or equal to VL, or greater than VU, will not be
55 returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.
56 IL (input) INTEGER
58 IU (input) INTEGER If RANGE='I', the indices (in ascending
59 order) of the smallest and largest eigenvalues to be returned.
60 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
61 referenced if RANGE = 'A' or 'V'.
62 ABSTOL (input) REAL
64 The absolute tolerance for the eigenvalues. An eigenvalue (or
65 cluster) is considered to be located if it has been determined
66 to lie in an interval whose width is ABSTOL or less. If ABSTOL
67 is less than or equal to zero, then ULP*|T| will be used, where
68 |T| means the 1-norm of T. Eigenvalues will be computed most
69 accurately when ABSTOL is set to twice the underflow threshold
70 2*SLAMCH('S'), not zero.
71 D (input) REAL array, dimension (N)
73 The n diagonal elements of the tridiagonal matrix T.
74 E (input) REAL array, dimension (N-1)
76 The (n-1) off-diagonal elements of the tridiagonal matrix T.
77 M (output) INTEGER
79 The actual number of eigenvalues found. 0 <= M <= N. (See also
80 the description of INFO=2,3.)
81 NSPLIT (output) INTEGER
83 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
84 N.
85 W (output) REAL array, dimension (N)
87 On exit, the first M elements of W will contain the eigenval‐
88 ues. (SSTEBZ may use the remaining N-M elements as workspace.)
89 IBLOCK (output) INTEGER array, dimension (N)
91 At each row/column j where E(j) is zero or small, the matrix T
92 is considered to split into a block diagonal matrix. On exit,
93 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
94 number of blocks) the eigenvalue W(i) belongs. (SSTEBZ may use
95 the remaining N-M elements as workspace.)
96 ISPLIT (output) 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 WORK (workspace) REAL array, dimension (4*N)
107 IWORK (workspace) INTEGER array, dimension (3*N)
109 INFO (output) INTEGER
111 = 0: successful exit
112 < 0: if INFO = -i, the i-th argument had an illegal value
113 > 0: some or all of the eigenvalues failed to converge or
114 were not computed:
115 =1 or 3: Bisection failed to converge for some eigenvalues;
116 these eigenvalues are flagged by a negative block number. The
117 effect is that the eigenvalues may not be as accurate as the
118 absolute and relative tolerances. This is generally caused by
119 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
120 Not all of the eigenvalues
121 IL:IU were found.
122 Effect: M < IU+1-IL
123 Cause: non-monotonic arithmetic, causing the Sturm sequence to
124 be non-monotonic. Cure: recalculate, using RANGE='A', and
125 pick
126 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
127 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
128 shgorin interval initially used was too small. No eigenvalues
129 were computed. Probable cause: your machine has sloppy float‐
130 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
131 recompile, and try again.
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134 RELFAC REAL, default = 2.0e0
135 The relative tolerance. An interval (a,b] lies within "rela‐
136 tive tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp"
137 is the machine precision (distance from 1 to the next larger
138 floating point number.)
139 FUDGE REAL, default = 2
141 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
142 value of 1 should work, but on machines with sloppy arithmetic,
143 this needs to be larger. The default for publicly released
144 versions should be large enough to handle the worst machine
145 around. Note that this has no effect on accuracy of the solu‐
146 tion.
147 LAPACK routine (version 3.2) November 2008 SSTEBZ(1)