1ZPTSVX(1) LAPACK routine (version 3.2) ZPTSVX(1)
2
6 ZPTSVX - 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 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. Error bounds on the solution and a condition estimate are
31 also provided.
32
34 The following steps are performed:
35 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
36 is a unit lower bidiagonal matrix and D is diagonal. The
37 factorization can also be regarded as having the form
38 A = U**H*D*U.
39 2. If the leading i-by-i principal minor is not positive definite,
40 then the routine returns with INFO = i. Otherwise, the factored
41 form of A is used to estimate the condition number of the matrix
42 A. If the reciprocal of the condition number is less than machine
43 precision, INFO = N+1 is returned as a warning, but the routine
44 still goes on to solve for X and compute error bounds as
45 described below.
46 3. The system of equations is solved for X using the factored form
47 of A.
48 4. Iterative refinement is applied to improve the computed solution
49 matrix and calculate error bounds and backward error estimates
50 for it.
51
53 FACT (input) CHARACTER*1
54 Specifies whether or not the factored form of the matrix A is
55 supplied on entry. = 'F': On entry, DF and EF contain the
56 factored form of A. D, E, DF, and EF will not be modified. =
57 'N': The matrix A will be copied to DF and EF and factored.
58 N (input) INTEGER
60 The order of the matrix A. N >= 0.
61 NRHS (input) INTEGER
63 The number of right hand sides, i.e., the number of columns of
64 the matrices B and X. NRHS >= 0.
65 D (input) DOUBLE PRECISION array, dimension (N)
67 The n diagonal elements of the tridiagonal matrix A.
68 E (input) COMPLEX*16 array, dimension (N-1)
70 The (n-1) subdiagonal elements of the tridiagonal matrix A.
71 DF (input or output) DOUBLE PRECISION array, dimension (N)
73 If FACT = 'F', then DF is an input argument and on entry con‐
74 tains the n diagonal elements of the diagonal matrix D from the
75 L*D*L**H factorization of A. If FACT = 'N', then DF is an out‐
76 put argument and on exit contains the n diagonal elements of
77 the diagonal matrix D from the L*D*L**H factorization of A.
78 EF (input or output) COMPLEX*16 array, dimension (N-1)
80 If FACT = 'F', then EF is an input argument and on entry con‐
81 tains the (n-1) subdiagonal elements of the unit bidiagonal
82 factor L from the L*D*L**H factorization of A. If FACT = 'N',
83 then EF is an output argument and on exit contains the (n-1)
84 subdiagonal elements of the unit bidiagonal factor L from the
85 L*D*L**H factorization of A.
86 B (input) COMPLEX*16 array, dimension (LDB,NRHS)
88 The N-by-NRHS right hand side matrix B.
89 LDB (input) INTEGER
91 The leading dimension of the array B. LDB >= max(1,N).
92 X (output) COMPLEX*16 array, dimension (LDX,NRHS)
94 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
95 LDX (input) INTEGER
97 The leading dimension of the array X. LDX >= max(1,N).
98 RCOND (output) DOUBLE PRECISION
100 The reciprocal condition number of the matrix A. If RCOND is
101 less than the machine precision (in particular, if RCOND = 0),
102 the matrix is singular to working precision. This condition is
103 indicated by a return code of INFO > 0.
104 FERR (output) DOUBLE PRECISION array, dimension (NRHS)
106 The forward error bound for each solution vector X(j) (the j-th
107 column of the solution matrix X). If XTRUE is the true solu‐
108 tion corresponding to X(j), FERR(j) is an estimated upper bound
109 for the magnitude of the largest element in (X(j) - XTRUE)
110 divided by the magnitude of the largest element in X(j).
111 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
113 The componentwise relative backward error of each solution vec‐
114 tor X(j) (i.e., the smallest relative change in any element of
115 A or B that makes X(j) an exact solution).
116 WORK (workspace) COMPLEX*16 array, dimension (N)
118 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
120 INFO (output) INTEGER
122 = 0: successful exit
123 < 0: if INFO = -i, the i-th argument had an illegal value
124 > 0: if INFO = i, and i is
125 <= N: the leading minor of order i of A is not positive defi‐
126 nite, so the factorization could not be completed, and the
127 solution has not been computed. RCOND = 0 is returned. = N+1:
128 U is nonsingular, but RCOND is less than machine precision,
129 meaning that the matrix is singular to working precision. Nev‐
130 ertheless, the solution and error bounds are computed because
131 there are a number of situations where the computed solution
132 can be more accurate than the value of RCOND would suggest.
133 LAPACK routine (version 3.2) November 2008 ZPTSVX(1)