1ZPTSVX(1) LAPACK routine (version 3.1) ZPTSVX(1)
2
6 ZPTSVX - 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 ZPTSVX( 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 DOUBLE PRECISION RCOND
20 DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
22 RWORK( * )
23 COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )
25
27 ZPTSVX uses the factorization A = L*D*L**H to compute the solution to a
28 complex system of linear equations A*X = B, where A is an N-by-N Her‐
29 mitian positive definite tridiagonal matrix and X and B are N-by-NRHS
30 matrices.
31 Error bounds on the solution and a condition estimate are also pro‐
33 vided.
34
37 The following steps are performed:
38 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
40 is a unit lower bidiagonal matrix and D is diagonal. The
41 factorization can also be regarded as having the form
42 A = U**H*D*U.
43 2. If the leading i-by-i principal minor is not positive definite,
45 then the routine returns with INFO = i. Otherwise, the factored
46 form of A is used to estimate the condition number of the matrix
47 A. If the reciprocal of the condition number is less than machine
48 precision, INFO = N+1 is returned as a warning, but the routine
49 still goes on to solve for X and compute error bounds as
50 described below.
51 3. The system of equations is solved for X using the factored form
53 of A.
54 4. Iterative refinement is applied to improve the computed solution
56 matrix and calculate error bounds and backward error estimates
57 for it.
58
61 FACT (input) CHARACTER*1
62 Specifies whether or not the factored form of the matrix A is
63 supplied on entry. = 'F': On entry, DF and EF contain the
64 factored form of A. D, E, DF, and EF will not be modified. =
65 'N': The matrix A will be copied to DF and EF and factored.
66 N (input) INTEGER
68 The order of the matrix A. N >= 0.
69 NRHS (input) INTEGER
71 The number of right hand sides, i.e., the number of columns of
72 the matrices B and X. NRHS >= 0.
73 D (input) DOUBLE PRECISION array, dimension (N)
75 The n diagonal elements of the tridiagonal matrix A.
76 E (input) COMPLEX*16 array, dimension (N-1)
78 The (n-1) subdiagonal elements of the tridiagonal matrix A.
79 DF (input or output) DOUBLE PRECISION array, dimension (N)
81 If FACT = 'F', then DF is an input argument and on entry con‐
82 tains the n diagonal elements of the diagonal matrix D from the
83 L*D*L**H factorization of A. If FACT = 'N', then DF is an out‐
84 put argument and on exit contains the n diagonal elements of
85 the diagonal matrix D from the L*D*L**H factorization of A.
86 EF (input or output) COMPLEX*16 array, dimension (N-1)
88 If FACT = 'F', then EF is an input argument and on entry con‐
89 tains the (n-1) subdiagonal elements of the unit bidiagonal
90 factor L from the L*D*L**H factorization of A. If FACT = 'N',
91 then EF is an output argument and on exit contains the (n-1)
92 subdiagonal elements of the unit bidiagonal factor L from the
93 L*D*L**H factorization of A.
94 B (input) COMPLEX*16 array, dimension (LDB,NRHS)
96 The N-by-NRHS right hand side matrix B.
97 LDB (input) INTEGER
99 The leading dimension of the array B. LDB >= max(1,N).
100 X (output) COMPLEX*16 array, dimension (LDX,NRHS)
102 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
103 LDX (input) INTEGER
105 The leading dimension of the array X. LDX >= max(1,N).
106 RCOND (output) DOUBLE PRECISION
108 The reciprocal condition number of the matrix A. If RCOND is
109 less than the machine precision (in particular, if RCOND = 0),
110 the matrix is singular to working precision. This condition is
111 indicated by a return code of INFO > 0.
112 FERR (output) DOUBLE PRECISION array, dimension (NRHS)
114 The forward error bound for each solution vector X(j) (the j-th
115 column of the solution matrix X). If XTRUE is the true solu‐
116 tion corresponding to X(j), FERR(j) is an estimated upper bound
117 for the magnitude of the largest element in (X(j) - XTRUE)
118 divided by the magnitude of the largest element in X(j).
119 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
121 The componentwise relative backward error of each solution vec‐
122 tor X(j) (i.e., the smallest relative change in any element of
123 A or B that makes X(j) an exact solution).
124 WORK (workspace) COMPLEX*16 array, dimension (N)
126 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
128 INFO (output) INTEGER
130 = 0: successful exit
131 < 0: if INFO = -i, the i-th argument had an illegal value
132 > 0: if INFO = i, and i is
133 <= N: the leading minor of order i of A is not positive defi‐
134 nite, so the factorization could not be completed, and the
135 solution has not been computed. RCOND = 0 is returned. = N+1:
136 U is nonsingular, but RCOND is less than machine precision,
137 meaning that the matrix is singular to working precision. Nev‐
138 ertheless, the solution and error bounds are computed because
139 there are a number of situations where the computed solution
140 can be more accurate than the value of RCOND would suggest.
141 LAPACK routine (version 3.1) November 2006 ZPTSVX(1)