1DSTEBZ(1) LAPACK routine (version 3.1) DSTEBZ(1)
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6 DSTEBZ - 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
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.
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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) DOUBLE PRECISION
55 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
56 upper bounds of the interval to be searched for eigenvalues.
57 Eigenvalues less than or equal to VL, or greater than VU, will
58 not be returned. VL < VU. Not referenced if RANGE = 'A' or
59 'I'.
60 IL (input) INTEGER
62 IU (input) INTEGER If RANGE='I', the indices (in ascending
63 order) of the smallest and largest eigenvalues to be returned.
64 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
65 referenced if RANGE = 'A' or 'V'.
66 ABSTOL (input) DOUBLE PRECISION
68 The absolute tolerance for the eigenvalues. An eigenvalue (or
69 cluster) is considered to be located if it has been determined
70 to lie in an interval whose width is ABSTOL or less. If ABSTOL
71 is less than or equal to zero, then ULP*|T| will be used, where
72 |T| means the 1-norm of T.
73 Eigenvalues will be computed most accurately when ABSTOL is set
75 to twice the underflow threshold 2*DLAMCH('S'), not zero.
76 D (input) DOUBLE PRECISION array, dimension (N)
78 The n diagonal elements of the tridiagonal matrix T.
79 E (input) DOUBLE PRECISION array, dimension (N-1)
81 The (n-1) off-diagonal elements of the tridiagonal matrix T.
82 M (output) INTEGER
84 The actual number of eigenvalues found. 0 <= M <= N. (See also
85 the description of INFO=2,3.)
86 NSPLIT (output) INTEGER
88 The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
89 N.
90 W (output) DOUBLE PRECISION array, dimension (N)
92 On exit, the first M elements of W will contain the eigenval‐
93 ues. (DSTEBZ may use the remaining N-M elements as workspace.)
94 IBLOCK (output) INTEGER array, dimension (N)
96 At each row/column j where E(j) is zero or small, the matrix T
97 is considered to split into a block diagonal matrix. On exit,
98 if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the
99 number of blocks) the eigenvalue W(i) belongs. (DSTEBZ may use
100 the remaining N-M elements as workspace.)
101 ISPLIT (output) INTEGER array, dimension (N)
103 The splitting points, at which T breaks up into submatrices.
104 The first submatrix consists of rows/columns 1 to ISPLIT(1),
105 the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
106 and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
107 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will
108 actually be used, but since the user cannot know a priori what
109 value NSPLIT will have, N words must be reserved for ISPLIT.)
110 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
112 IWORK (workspace) INTEGER array, dimension (3*N)
114 INFO (output) INTEGER
116 = 0: successful exit
117 < 0: if INFO = -i, the i-th argument had an illegal value
118 > 0: some or all of the eigenvalues failed to converge or
119 were not computed:
120 =1 or 3: Bisection failed to converge for some eigenvalues;
121 these eigenvalues are flagged by a negative block number. The
122 effect is that the eigenvalues may not be as accurate as the
123 absolute and relative tolerances. This is generally caused by
124 unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only:
125 Not all of the eigenvalues
126 IL:IU were found.
127 Effect: M < IU+1-IL
128 Cause: non-monotonic arithmetic, causing the Sturm sequence to
129 be non-monotonic. Cure: recalculate, using RANGE='A', and
130 pick
131 out eigenvalues IL:IU. In some cases, increasing the PARAMETER
132 "FUDGE" may make things work. = 4: RANGE='I', and the Ger‐
133 shgorin interval initially used was too small. No eigenvalues
134 were computed. Probable cause: your machine has sloppy float‐
135 ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
136 recompile, and try again.
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139 RELFAC DOUBLE PRECISION, default = 2.0e0
140 The relative tolerance. An interval (a,b] lies within "rela‐
141 tive tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp"
142 is the machine precision (distance from 1 to the next larger
143 floating point number.)
144 FUDGE DOUBLE PRECISION, default = 2
146 A "fudge factor" to widen the Gershgorin intervals. Ideally, a
147 value of 1 should work, but on machines with sloppy arithmetic,
148 this needs to be larger. The default for publicly released
149 versions should be large enough to handle the worst machine
150 around. Note that this has no effect on accuracy of the solu‐
151 tion.
152 LAPACK routine (version 3.1) November 2006 DSTEBZ(1)