Annotation of rpl/lapack/lapack/zgtsvx.f, revision 1.5
1.1 bertrand 1: SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
2: $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
3: $ WORK, RWORK, INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER FACT, TRANS
12: INTEGER INFO, LDB, LDX, N, NRHS
13: DOUBLE PRECISION RCOND
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * )
17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18: COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
19: $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
20: $ WORK( * ), X( LDX, * )
21: * ..
22: *
23: * Purpose
24: * =======
25: *
26: * ZGTSVX uses the LU factorization to compute the solution to a complex
27: * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
28: * where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
29: * matrices.
30: *
31: * Error bounds on the solution and a condition estimate are also
32: * provided.
33: *
34: * Description
35: * ===========
36: *
37: * The following steps are performed:
38: *
39: * 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
40: * as A = L * U, where L is a product of permutation and unit lower
41: * bidiagonal matrices and U is upper triangular with nonzeros in
42: * only the main diagonal and first two superdiagonals.
43: *
44: * 2. If some U(i,i)=0, so that U is exactly singular, then the routine
45: * returns with INFO = i. Otherwise, the factored form of A is used
46: * to estimate the condition number of the matrix A. If the
47: * reciprocal of the condition number is less than machine precision,
48: * INFO = N+1 is returned as a warning, but the routine still goes on
49: * to solve for X and compute error bounds as described below.
50: *
51: * 3. The system of equations is solved for X using the factored form
52: * of A.
53: *
54: * 4. Iterative refinement is applied to improve the computed solution
55: * matrix and calculate error bounds and backward error estimates
56: * for it.
57: *
58: * Arguments
59: * =========
60: *
61: * FACT (input) CHARACTER*1
62: * Specifies whether or not the factored form of A has been
63: * supplied on entry.
64: * = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form
65: * of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
66: * be modified.
67: * = 'N': The matrix will be copied to DLF, DF, and DUF
68: * and factored.
69: *
70: * TRANS (input) CHARACTER*1
71: * Specifies the form of the system of equations:
72: * = 'N': A * X = B (No transpose)
73: * = 'T': A**T * X = B (Transpose)
74: * = 'C': A**H * X = B (Conjugate transpose)
75: *
76: * N (input) INTEGER
77: * The order of the matrix A. N >= 0.
78: *
79: * NRHS (input) INTEGER
80: * The number of right hand sides, i.e., the number of columns
81: * of the matrix B. NRHS >= 0.
82: *
83: * DL (input) COMPLEX*16 array, dimension (N-1)
84: * The (n-1) subdiagonal elements of A.
85: *
86: * D (input) COMPLEX*16 array, dimension (N)
87: * The n diagonal elements of A.
88: *
89: * DU (input) COMPLEX*16 array, dimension (N-1)
90: * The (n-1) superdiagonal elements of A.
91: *
92: * DLF (input or output) COMPLEX*16 array, dimension (N-1)
93: * If FACT = 'F', then DLF is an input argument and on entry
94: * contains the (n-1) multipliers that define the matrix L from
95: * the LU factorization of A as computed by ZGTTRF.
96: *
97: * If FACT = 'N', then DLF is an output argument and on exit
98: * contains the (n-1) multipliers that define the matrix L from
99: * the LU factorization of A.
100: *
101: * DF (input or output) COMPLEX*16 array, dimension (N)
102: * If FACT = 'F', then DF is an input argument and on entry
103: * contains the n diagonal elements of the upper triangular
104: * matrix U from the LU factorization of A.
105: *
106: * If FACT = 'N', then DF is an output argument and on exit
107: * contains the n diagonal elements of the upper triangular
108: * matrix U from the LU factorization of A.
109: *
110: * DUF (input or output) COMPLEX*16 array, dimension (N-1)
111: * If FACT = 'F', then DUF is an input argument and on entry
112: * contains the (n-1) elements of the first superdiagonal of U.
113: *
114: * If FACT = 'N', then DUF is an output argument and on exit
115: * contains the (n-1) elements of the first superdiagonal of U.
116: *
117: * DU2 (input or output) COMPLEX*16 array, dimension (N-2)
118: * If FACT = 'F', then DU2 is an input argument and on entry
119: * contains the (n-2) elements of the second superdiagonal of
120: * U.
121: *
122: * If FACT = 'N', then DU2 is an output argument and on exit
123: * contains the (n-2) elements of the second superdiagonal of
124: * U.
125: *
126: * IPIV (input or output) INTEGER array, dimension (N)
127: * If FACT = 'F', then IPIV is an input argument and on entry
128: * contains the pivot indices from the LU factorization of A as
129: * computed by ZGTTRF.
130: *
131: * If FACT = 'N', then IPIV is an output argument and on exit
132: * contains the pivot indices from the LU factorization of A;
133: * row i of the matrix was interchanged with row IPIV(i).
134: * IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
135: * a row interchange was not required.
136: *
137: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
138: * The N-by-NRHS right hand side matrix B.
139: *
140: * LDB (input) INTEGER
141: * The leading dimension of the array B. LDB >= max(1,N).
142: *
143: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
144: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
145: *
146: * LDX (input) INTEGER
147: * The leading dimension of the array X. LDX >= max(1,N).
148: *
149: * RCOND (output) DOUBLE PRECISION
150: * The estimate of the reciprocal condition number of the matrix
151: * A. If RCOND is less than the machine precision (in
152: * particular, if RCOND = 0), the matrix is singular to working
153: * precision. This condition is indicated by a return code of
154: * INFO > 0.
155: *
156: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
157: * The estimated forward error bound for each solution vector
158: * X(j) (the j-th column of the solution matrix X).
159: * If XTRUE is the true solution corresponding to X(j), FERR(j)
160: * is an estimated upper bound for the magnitude of the largest
161: * element in (X(j) - XTRUE) divided by the magnitude of the
162: * largest element in X(j). The estimate is as reliable as
163: * the estimate for RCOND, and is almost always a slight
164: * overestimate of the true error.
165: *
166: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
167: * The componentwise relative backward error of each solution
168: * vector X(j) (i.e., the smallest relative change in
169: * any element of A or B that makes X(j) an exact solution).
170: *
171: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
172: *
173: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
174: *
175: * INFO (output) INTEGER
176: * = 0: successful exit
177: * < 0: if INFO = -i, the i-th argument had an illegal value
178: * > 0: if INFO = i, and i is
179: * <= N: U(i,i) is exactly zero. The factorization
180: * has not been completed unless i = N, but the
181: * factor U is exactly singular, so the solution
182: * and error bounds could not be computed.
183: * RCOND = 0 is returned.
184: * = N+1: U is nonsingular, but RCOND is less than machine
185: * precision, meaning that the matrix is singular
186: * to working precision. Nevertheless, the
187: * solution and error bounds are computed because
188: * there are a number of situations where the
189: * computed solution can be more accurate than the
190: * value of RCOND would suggest.
191: *
192: * =====================================================================
193: *
194: * .. Parameters ..
195: DOUBLE PRECISION ZERO
196: PARAMETER ( ZERO = 0.0D+0 )
197: * ..
198: * .. Local Scalars ..
199: LOGICAL NOFACT, NOTRAN
200: CHARACTER NORM
201: DOUBLE PRECISION ANORM
202: * ..
203: * .. External Functions ..
204: LOGICAL LSAME
205: DOUBLE PRECISION DLAMCH, ZLANGT
206: EXTERNAL LSAME, DLAMCH, ZLANGT
207: * ..
208: * .. External Subroutines ..
209: EXTERNAL XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS,
210: $ ZLACPY
211: * ..
212: * .. Intrinsic Functions ..
213: INTRINSIC MAX
214: * ..
215: * .. Executable Statements ..
216: *
217: INFO = 0
218: NOFACT = LSAME( FACT, 'N' )
219: NOTRAN = LSAME( TRANS, 'N' )
220: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
221: INFO = -1
222: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
223: $ LSAME( TRANS, 'C' ) ) THEN
224: INFO = -2
225: ELSE IF( N.LT.0 ) THEN
226: INFO = -3
227: ELSE IF( NRHS.LT.0 ) THEN
228: INFO = -4
229: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
230: INFO = -14
231: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
232: INFO = -16
233: END IF
234: IF( INFO.NE.0 ) THEN
235: CALL XERBLA( 'ZGTSVX', -INFO )
236: RETURN
237: END IF
238: *
239: IF( NOFACT ) THEN
240: *
241: * Compute the LU factorization of A.
242: *
243: CALL ZCOPY( N, D, 1, DF, 1 )
244: IF( N.GT.1 ) THEN
245: CALL ZCOPY( N-1, DL, 1, DLF, 1 )
246: CALL ZCOPY( N-1, DU, 1, DUF, 1 )
247: END IF
248: CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
249: *
250: * Return if INFO is non-zero.
251: *
252: IF( INFO.GT.0 )THEN
253: RCOND = ZERO
254: RETURN
255: END IF
256: END IF
257: *
258: * Compute the norm of the matrix A.
259: *
260: IF( NOTRAN ) THEN
261: NORM = '1'
262: ELSE
263: NORM = 'I'
264: END IF
265: ANORM = ZLANGT( NORM, N, DL, D, DU )
266: *
267: * Compute the reciprocal of the condition number of A.
268: *
269: CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
270: $ INFO )
271: *
272: * Compute the solution vectors X.
273: *
274: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
275: CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
276: $ INFO )
277: *
278: * Use iterative refinement to improve the computed solutions and
279: * compute error bounds and backward error estimates for them.
280: *
281: CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
282: $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
283: *
284: * Set INFO = N+1 if the matrix is singular to working precision.
285: *
286: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
287: $ INFO = N + 1
288: *
289: RETURN
290: *
291: * End of ZGTSVX
292: *
293: END
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