1CPTSVX(1) LAPACK routine (version 3.2) CPTSVX(1)
2
6 CPTSVX - uses the factorization A = L*D*L**H to compute the solution to
7 a complex system of linear equations A*X = B, where A is an N-by-N Her‐
8 mitian positive definite tridiagonal matrix and X and B are N-by-NRHS
9 matrices
10
12 SUBROUTINE CPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND,
13 FERR, BERR, WORK, RWORK, INFO )
14 CHARACTER FACT
16 INTEGER INFO, LDB, LDX, N, NRHS
18 REAL RCOND
20 REAL BERR( * ), D( * ), DF( * ), FERR( * ), RWORK( * )
22 COMPLEX B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )
24
26 CPTSVX uses the factorization A = L*D*L**H to compute the solution to a
27 complex system of linear equations A*X = B, where A is an N-by-N Her‐
28 mitian positive definite tridiagonal matrix and X and B are N-by-NRHS
29 matrices. Error bounds on the solution and a condition estimate are
30 also provided.
31
33 The following steps are performed:
34 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
35 is a unit lower bidiagonal matrix and D is diagonal. The
36 factorization can also be regarded as having the form
37 A = U**H*D*U.
38 2. If the leading i-by-i principal minor is not positive definite,
39 then the routine returns with INFO = i. Otherwise, the factored
40 form of A is used to estimate the condition number of the matrix
41 A. If the reciprocal of the condition number is less than machine
42 precision, INFO = N+1 is returned as a warning, but the routine
43 still goes on to solve for X and compute error bounds as
44 described below.
45 3. The system of equations is solved for X using the factored form
46 of A.
47 4. Iterative refinement is applied to improve the computed solution
48 matrix and calculate error bounds and backward error estimates
49 for it.
50
52 FACT (input) CHARACTER*1
53 Specifies whether or not the factored form of the matrix A is
54 supplied on entry. = 'F': On entry, DF and EF contain the
55 factored form of A. D, E, DF, and EF will not be modified. =
56 'N': The matrix A will be copied to DF and EF and factored.
57 N (input) INTEGER
59 The order of the matrix A. N >= 0.
60 NRHS (input) INTEGER
62 The number of right hand sides, i.e., the number of columns of
63 the matrices B and X. NRHS >= 0.
64 D (input) REAL array, dimension (N)
66 The n diagonal elements of the tridiagonal matrix A.
67 E (input) COMPLEX array, dimension (N-1)
69 The (n-1) subdiagonal elements of the tridiagonal matrix A.
70 DF (input or output) REAL array, dimension (N)
72 If FACT = 'F', then DF is an input argument and on entry con‐
73 tains the n diagonal elements of the diagonal matrix D from the
74 L*D*L**H factorization of A. If FACT = 'N', then DF is an out‐
75 put argument and on exit contains the n diagonal elements of
76 the diagonal matrix D from the L*D*L**H factorization of A.
77 EF (input or output) COMPLEX array, dimension (N-1)
79 If FACT = 'F', then EF is an input argument and on entry con‐
80 tains the (n-1) subdiagonal elements of the unit bidiagonal
81 factor L from the L*D*L**H factorization of A. If FACT = 'N',
82 then EF is an output argument and on exit contains the (n-1)
83 subdiagonal elements of the unit bidiagonal factor L from the
84 L*D*L**H factorization of A.
85 B (input) COMPLEX array, dimension (LDB,NRHS)
87 The N-by-NRHS right hand side matrix B.
88 LDB (input) INTEGER
90 The leading dimension of the array B. LDB >= max(1,N).
91 X (output) COMPLEX array, dimension (LDX,NRHS)
93 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
94 LDX (input) INTEGER
96 The leading dimension of the array X. LDX >= max(1,N).
97 RCOND (output) REAL
99 The reciprocal condition number of the matrix A. If RCOND is
100 less than the machine precision (in particular, if RCOND = 0),
101 the matrix is singular to working precision. This condition is
102 indicated by a return code of INFO > 0.
103 FERR (output) REAL array, dimension (NRHS)
105 The forward error bound for each solution vector X(j) (the j-th
106 column of the solution matrix X). If XTRUE is the true solu‐
107 tion corresponding to X(j), FERR(j) is an estimated upper bound
108 for the magnitude of the largest element in (X(j) - XTRUE)
109 divided by the magnitude of the largest element in X(j).
110 BERR (output) REAL array, dimension (NRHS)
112 The componentwise relative backward error of each solution vec‐
113 tor X(j) (i.e., the smallest relative change in any element of
114 A or B that makes X(j) an exact solution).
115 WORK (workspace) COMPLEX array, dimension (N)
117 RWORK (workspace) REAL array, dimension (N)
119 INFO (output) INTEGER
121 = 0: successful exit
122 < 0: if INFO = -i, the i-th argument had an illegal value
123 > 0: if INFO = i, and i is
124 <= N: the leading minor of order i of A is not positive defi‐
125 nite, so the factorization could not be completed, and the
126 solution has not been computed. RCOND = 0 is returned. = N+1:
127 U is nonsingular, but RCOND is less than machine precision,
128 meaning that the matrix is singular to working precision. Nev‐
129 ertheless, the solution and error bounds are computed because
130 there are a number of situations where the computed solution
131 can be more accurate than the value of RCOND would suggest.
132 LAPACK routine (version 3.2) November 2008 CPTSVX(1)