1DLALN2(1) LAPACK auxiliary routine (version 3.2) DLALN2(1)
2
6 DLALN2 - solves a system of the form (ca A - w D ) X = s B or (ca A' -
7 w D) X = s B with possible scaling ("s") and perturbation of A
8
10 SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB,
11 WR, WI, X, LDX, SCALE, XNORM, INFO )
12 LOGICAL LTRANS
14 INTEGER INFO, LDA, LDB, LDX, NA, NW
16 DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
18 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
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22 DLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A' - w
23 D) X = s B with possible scaling ("s") and perturbation of A. (A'
24 means A-transpose.) A is an NA x NA real matrix, ca is a real scalar,
25 D is an NA x NA real diagonal matrix, w is a real or complex value, and
26 X and B are NA x 1 matrices -- real if w is real, complex if w is com‐
27 plex. NA may be 1 or 2.
28 If w is complex, X and B are represented as NA x 2 matrices, the first
29 column of each being the real part and the second being the imaginary
30 part.
31 "s" is a scaling factor (.LE. 1), computed by DLALN2, which is so cho‐
32 sen that X can be computed without overflow. X is further scaled if
33 necessary to assure that norm(ca A - w D)*norm(X) is less than over‐
34 flow.
35 If both singular values of (ca A - w D) are less than SMIN, SMIN*iden‐
36 tity will be used instead of (ca A - w D). If only one singular value
37 is less than SMIN, one element of (ca A - w D) will be perturbed enough
38 to make the smallest singular value roughly SMIN. If both singular
39 values are at least SMIN, (ca A - w D) will not be perturbed. In any
40 case, the perturbation will be at most some small multiple of max(
41 SMIN, ulp*norm(ca A - w D) ). The singular values are computed by
42 infinity-norm approximations, and thus will only be correct to a factor
43 of 2 or so.
44 Note: all input quantities are assumed to be smaller than overflow by a
45 reasonable factor. (See BIGNUM.)
46
48 LTRANS (input) LOGICAL
49 =.TRUE.: A-transpose will be used.
50 =.FALSE.: A will be used (not transposed.)
51 NA (input) INTEGER
53 The size of the matrix A. It may (only) be 1 or 2.
54 NW (input) INTEGER
56 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.
57 SMIN (input) DOUBLE PRECISION
59 The desired lower bound on the singular values of A. This
60 should be a safe distance away from underflow or overflow, say,
61 between (underflow/machine precision) and (machine precision *
62 overflow ). (See BIGNUM and ULP.)
63 CA (input) DOUBLE PRECISION
65 The coefficient c, which A is multiplied by.
66 A (input) DOUBLE PRECISION array, dimension (LDA,NA)
68 The NA x NA matrix A.
69 LDA (input) INTEGER
71 The leading dimension of A. It must be at least NA.
72 D1 (input) DOUBLE PRECISION
74 The 1,1 element in the diagonal matrix D.
75 D2 (input) DOUBLE PRECISION
77 The 2,2 element in the diagonal matrix D. Not used if NW=1.
78 B (input) DOUBLE PRECISION array, dimension (LDB,NW)
80 The NA x NW matrix B (right-hand side). If NW=2 ("w" is com‐
81 plex), column 1 contains the real part of B and column 2 con‐
82 tains the imaginary part.
83 LDB (input) INTEGER
85 The leading dimension of B. It must be at least NA.
86 WR (input) DOUBLE PRECISION
88 The real part of the scalar "w".
89 WI (input) DOUBLE PRECISION
91 The imaginary part of the scalar "w". Not used if NW=1.
92 X (output) DOUBLE PRECISION array, dimension (LDX,NW)
94 The NA x NW matrix X (unknowns), as computed by DLALN2. If
95 NW=2 ("w" is complex), on exit, column 1 will contain the real
96 part of X and column 2 will contain the imaginary part.
97 LDX (input) INTEGER
99 The leading dimension of X. It must be at least NA.
100 SCALE (output) DOUBLE PRECISION
102 The scale factor that B must be multiplied by to insure that
103 overflow does not occur when computing X. Thus, (ca A - w D) X
104 will be SCALE*B, not B (ignoring perturbations of A.) It will
105 be at most 1.
106 XNORM (output) DOUBLE PRECISION
108 The infinity-norm of X, when X is regarded as an NA x NW real
109 matrix.
110 INFO (output) INTEGER
112 An error flag. It will be set to zero if no error occurs, a
113 negative number if an argument is in error, or a positive num‐
114 ber if ca A - w D had to be perturbed. The possible values
115 are:
116 = 0: No error occurred, and (ca A - w D) did not have to be
117 perturbed. = 1: (ca A - w D) had to be perturbed to make its
118 smallest (or only) singular value greater than SMIN. NOTE: In
119 the interests of speed, this routine does not check the inputs
120 for errors.
121 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLALN2(1)