1CPTSVX(1) LAPACK routine (version 3.1) CPTSVX(1)
2
6 CPTSVX - the factorization A = L*D*L**H to compute the solution to a
7 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.
30 Error bounds on the solution and a condition estimate are also pro‐
32 vided.
33
36 The following steps are performed:
37 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
39 is a unit lower bidiagonal matrix and D is diagonal. The
40 factorization can also be regarded as having the form
41 A = U**H*D*U.
42 2. If the leading i-by-i principal minor is not positive definite,
44 then the routine returns with INFO = i. Otherwise, the factored
45 form of A is used to estimate the condition number of the matrix
46 A. If the reciprocal of the condition number is less than machine
47 precision, INFO = N+1 is returned as a warning, but the routine
48 still goes on to solve for X and compute error bounds as
49 described below.
50 3. The system of equations is solved for X using the factored form
52 of A.
53 4. Iterative refinement is applied to improve the computed solution
55 matrix and calculate error bounds and backward error estimates
56 for it.
57
60 FACT (input) CHARACTER*1
61 Specifies whether or not the factored form of the matrix A is
62 supplied on entry. = 'F': On entry, DF and EF contain the
63 factored form of A. D, E, DF, and EF will not be modified. =
64 'N': The matrix A will be copied to DF and EF and factored.
65 N (input) INTEGER
67 The order of the matrix A. N >= 0.
68 NRHS (input) INTEGER
70 The number of right hand sides, i.e., the number of columns of
71 the matrices B and X. NRHS >= 0.
72 D (input) REAL array, dimension (N)
74 The n diagonal elements of the tridiagonal matrix A.
75 E (input) COMPLEX array, dimension (N-1)
77 The (n-1) subdiagonal elements of the tridiagonal matrix A.
78 DF (input or output) REAL array, dimension (N)
80 If FACT = 'F', then DF is an input argument and on entry con‐
81 tains the n diagonal elements of the diagonal matrix D from the
82 L*D*L**H factorization of A. If FACT = 'N', then DF is an out‐
83 put argument and on exit contains the n diagonal elements of
84 the diagonal matrix D from the L*D*L**H factorization of A.
85 EF (input or output) COMPLEX array, dimension (N-1)
87 If FACT = 'F', then EF is an input argument and on entry con‐
88 tains the (n-1) subdiagonal elements of the unit bidiagonal
89 factor L from the L*D*L**H factorization of A. If FACT = 'N',
90 then EF is an output argument and on exit contains the (n-1)
91 subdiagonal elements of the unit bidiagonal factor L from the
92 L*D*L**H factorization of A.
93 B (input) COMPLEX array, dimension (LDB,NRHS)
95 The N-by-NRHS right hand side matrix B.
96 LDB (input) INTEGER
98 The leading dimension of the array B. LDB >= max(1,N).
99 X (output) COMPLEX array, dimension (LDX,NRHS)
101 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
102 LDX (input) INTEGER
104 The leading dimension of the array X. LDX >= max(1,N).
105 RCOND (output) REAL
107 The reciprocal condition number of the matrix A. If RCOND is
108 less than the machine precision (in particular, if RCOND = 0),
109 the matrix is singular to working precision. This condition is
110 indicated by a return code of INFO > 0.
111 FERR (output) REAL array, dimension (NRHS)
113 The forward error bound for each solution vector X(j) (the j-th
114 column of the solution matrix X). If XTRUE is the true solu‐
115 tion corresponding to X(j), FERR(j) is an estimated upper bound
116 for the magnitude of the largest element in (X(j) - XTRUE)
117 divided by the magnitude of the largest element in X(j).
118 BERR (output) REAL array, dimension (NRHS)
120 The componentwise relative backward error of each solution vec‐
121 tor X(j) (i.e., the smallest relative change in any element of
122 A or B that makes X(j) an exact solution).
123 WORK (workspace) COMPLEX array, dimension (N)
125 RWORK (workspace) REAL array, dimension (N)
127 INFO (output) INTEGER
129 = 0: successful exit
130 < 0: if INFO = -i, the i-th argument had an illegal value
131 > 0: if INFO = i, and i is
132 <= N: the leading minor of order i of A is not positive defi‐
133 nite, so the factorization could not be completed, and the
134 solution has not been computed. RCOND = 0 is returned. = N+1:
135 U is nonsingular, but RCOND is less than machine precision,
136 meaning that the matrix is singular to working precision. Nev‐
137 ertheless, the solution and error bounds are computed because
138 there are a number of situations where the computed solution
139 can be more accurate than the value of RCOND would suggest.
140 LAPACK routine (version 3.1) November 2006 CPTSVX(1)