1DLALN2(1) LAPACK auxiliary routine (version 3.2) DLALN2(1)
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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
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10 SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB,
11 WR, WI, X, LDX, SCALE, XNORM, INFO )
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13 LOGICAL LTRANS
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15 INTEGER INFO, LDA, LDB, LDX, NA, NW
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17 DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
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19 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.)
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48 LTRANS (input) LOGICAL
49 =.TRUE.: A-transpose will be used.
50 =.FALSE.: A will be used (not transposed.)
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52 NA (input) INTEGER
53 The size of the matrix A. It may (only) be 1 or 2.
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55 NW (input) INTEGER
56 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2.
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58 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.)
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64 CA (input) DOUBLE PRECISION
65 The coefficient c, which A is multiplied by.
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67 A (input) DOUBLE PRECISION array, dimension (LDA,NA)
68 The NA x NA matrix A.
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70 LDA (input) INTEGER
71 The leading dimension of A. It must be at least NA.
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73 D1 (input) DOUBLE PRECISION
74 The 1,1 element in the diagonal matrix D.
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76 D2 (input) DOUBLE PRECISION
77 The 2,2 element in the diagonal matrix D. Not used if NW=1.
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79 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.
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84 LDB (input) INTEGER
85 The leading dimension of B. It must be at least NA.
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87 WR (input) DOUBLE PRECISION
88 The real part of the scalar "w".
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90 WI (input) DOUBLE PRECISION
91 The imaginary part of the scalar "w". Not used if NW=1.
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93 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.
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98 LDX (input) INTEGER
99 The leading dimension of X. It must be at least NA.
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101 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.
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107 XNORM (output) DOUBLE PRECISION
108 The infinity-norm of X, when X is regarded as an NA x NW real
109 matrix.
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111 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
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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.
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124 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 DLALN2(1)