1DSTEBZ(1) LAPACK routine (version 3.2) DSTEBZ(1)
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6 DSTEBZ - computes the eigenvalues of a symmetric tridiagonal matrix T
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9 SUBROUTINE DSTEBZ( 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 DOUBLE PRECISION ABSTOL, VL, VU
17 INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
19 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
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23 DSTEBZ 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) DOUBLE PRECISION
52 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
53 upper bounds of the interval to be searched for eigenvalues.
54 Eigenvalues less than or equal to VL, or greater than VU, will
55 not be returned. VL < VU. Not referenced if RANGE = 'A' or
56 'I'.
57 IL (input) INTEGER
59 IU (input) INTEGER If RANGE='I', the indices (in ascending
60 order) of the smallest and largest eigenvalues to be returned.
61 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
62 referenced if RANGE = 'A' or 'V'.
63 ABSTOL (input) DOUBLE PRECISION
65 The absolute tolerance for the eigenvalues. An eigenvalue (or
66 cluster) is considered to be located if it has been determined
67 to lie in an interval whose width is ABSTOL or less. If ABSTOL
68 is less than or equal to zero, then ULP*|T| will be used, where
69 |T| means the 1-norm of T. Eigenvalues will be computed most
70 accurately when ABSTOL is set to twice the underflow threshold
71 2*DLAMCH('S'), not zero.
72 D (input) DOUBLE PRECISION array, dimension (N)
74 The n diagonal elements of the tridiagonal matrix T.
75 E (input) DOUBLE PRECISION array, dimension (N-1)
77 The (n-1) off-diagonal elements of the tridiagonal matrix T.
78 M (output) INTEGER
80 The actual number of eigenvalues found. 0 <= M <= N. (See also
81 the description of INFO=2,3.)
82 NSPLIT (output) INTEGER
84 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
85 N.
86 W (output) DOUBLE PRECISION array, dimension (N)
88 On exit, the first M elements of W will contain the eigenval‐
89 ues. (DSTEBZ may use the remaining N-M elements as workspace.)
90 IBLOCK (output) INTEGER array, dimension (N)
92 At each row/column j where E(j) is zero or small, the matrix T
93 is considered to split into a block diagonal matrix. On exit,
94 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
95 number of blocks) the eigenvalue W(i) belongs. (DSTEBZ may use
96 the remaining N-M elements as workspace.)
97 ISPLIT (output) INTEGER array, dimension (N)
99 The splitting points, at which T breaks up into submatrices.
100 The first submatrix consists of rows/columns 1 to ISPLIT(1),
101 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
102 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
103 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
104 actually be used, but since the user cannot know a priori what
105 value NSPLIT will have, N words must be reserved for ISPLIT.)
106 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
108 IWORK (workspace) INTEGER array, dimension (3*N)
110 INFO (output) INTEGER
112 = 0: successful exit
113 < 0: if INFO = -i, the i-th argument had an illegal value
114 > 0: some or all of the eigenvalues failed to converge or
115 were not computed:
116 =1 or 3: Bisection failed to converge for some eigenvalues;
117 these eigenvalues are flagged by a negative block number. The
118 effect is that the eigenvalues may not be as accurate as the
119 absolute and relative tolerances. This is generally caused by
120 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
121 Not all of the eigenvalues
122 IL:IU were found.
123 Effect: M < IU+1-IL
124 Cause: non-monotonic arithmetic, causing the Sturm sequence to
125 be non-monotonic. Cure: recalculate, using RANGE='A', and
126 pick
127 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
128 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
129 shgorin interval initially used was too small. No eigenvalues
130 were computed. Probable cause: your machine has sloppy float‐
131 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
132 recompile, and try again.
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135 RELFAC DOUBLE PRECISION, default = 2.0e0
136 The relative tolerance. An interval (a,b] lies within "rela‐
137 tive tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp"
138 is the machine precision (distance from 1 to the next larger
139 floating point number.)
140 FUDGE DOUBLE PRECISION, default = 2
142 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
143 value of 1 should work, but on machines with sloppy arithmetic,
144 this needs to be larger. The default for publicly released
145 versions should be large enough to handle the worst machine
146 around. Note that this has no effect on accuracy of the solu‐
147 tion.
148 LAPACK routine (version 3.2) November 2008 DSTEBZ(1)