1: *> \brief <b> DSGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement)
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
22: * SWORK, ITER, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * REAL SWORK( * )
30: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
31: * $ X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DSGESV computes the solution to a real system of linear equations
41: *> A * X = B,
42: *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
43: *>
44: *> DSGESV first attempts to factorize the matrix in SINGLE PRECISION
45: *> and use this factorization within an iterative refinement procedure
46: *> to produce a solution with DOUBLE PRECISION normwise backward error
47: *> quality (see below). If the approach fails the method switches to a
48: *> DOUBLE PRECISION factorization and solve.
49: *>
50: *> The iterative refinement is not going to be a winning strategy if
51: *> the ratio SINGLE PRECISION performance over DOUBLE PRECISION
52: *> performance is too small. A reasonable strategy should take the
53: *> number of right-hand sides and the size of the matrix into account.
54: *> This might be done with a call to ILAENV in the future. Up to now, we
55: *> always try iterative refinement.
56: *>
57: *> The iterative refinement process is stopped if
58: *> ITER > ITERMAX
59: *> or for all the RHS we have:
60: *> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
61: *> where
62: *> o ITER is the number of the current iteration in the iterative
63: *> refinement process
64: *> o RNRM is the infinity-norm of the residual
65: *> o XNRM is the infinity-norm of the solution
66: *> o ANRM is the infinity-operator-norm of the matrix A
67: *> o EPS is the machine epsilon returned by DLAMCH('Epsilon')
68: *> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
69: *> respectively.
70: *> \endverbatim
71: *
72: * Arguments:
73: * ==========
74: *
75: *> \param[in] N
76: *> \verbatim
77: *> N is INTEGER
78: *> The number of linear equations, i.e., the order of the
79: *> matrix A. N >= 0.
80: *> \endverbatim
81: *>
82: *> \param[in] NRHS
83: *> \verbatim
84: *> NRHS is INTEGER
85: *> The number of right hand sides, i.e., the number of columns
86: *> of the matrix B. NRHS >= 0.
87: *> \endverbatim
88: *>
89: *> \param[in,out] A
90: *> \verbatim
91: *> A is DOUBLE PRECISION array,
92: *> dimension (LDA,N)
93: *> On entry, the N-by-N coefficient matrix A.
94: *> On exit, if iterative refinement has been successfully used
95: *> (INFO.EQ.0 and ITER.GE.0, see description below), then A is
96: *> unchanged, if double precision factorization has been used
97: *> (INFO.EQ.0 and ITER.LT.0, see description below), then the
98: *> array A contains the factors L and U from the factorization
99: *> A = P*L*U; the unit diagonal elements of L are not stored.
100: *> \endverbatim
101: *>
102: *> \param[in] LDA
103: *> \verbatim
104: *> LDA is INTEGER
105: *> The leading dimension of the array A. LDA >= max(1,N).
106: *> \endverbatim
107: *>
108: *> \param[out] IPIV
109: *> \verbatim
110: *> IPIV is INTEGER array, dimension (N)
111: *> The pivot indices that define the permutation matrix P;
112: *> row i of the matrix was interchanged with row IPIV(i).
113: *> Corresponds either to the single precision factorization
114: *> (if INFO.EQ.0 and ITER.GE.0) or the double precision
115: *> factorization (if INFO.EQ.0 and ITER.LT.0).
116: *> \endverbatim
117: *>
118: *> \param[in] B
119: *> \verbatim
120: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
121: *> The N-by-NRHS right hand side matrix B.
122: *> \endverbatim
123: *>
124: *> \param[in] LDB
125: *> \verbatim
126: *> LDB is INTEGER
127: *> The leading dimension of the array B. LDB >= max(1,N).
128: *> \endverbatim
129: *>
130: *> \param[out] X
131: *> \verbatim
132: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
133: *> If INFO = 0, the N-by-NRHS solution matrix X.
134: *> \endverbatim
135: *>
136: *> \param[in] LDX
137: *> \verbatim
138: *> LDX is INTEGER
139: *> The leading dimension of the array X. LDX >= max(1,N).
140: *> \endverbatim
141: *>
142: *> \param[out] WORK
143: *> \verbatim
144: *> WORK is DOUBLE PRECISION array, dimension (N,NRHS)
145: *> This array is used to hold the residual vectors.
146: *> \endverbatim
147: *>
148: *> \param[out] SWORK
149: *> \verbatim
150: *> SWORK is REAL array, dimension (N*(N+NRHS))
151: *> This array is used to use the single precision matrix and the
152: *> right-hand sides or solutions in single precision.
153: *> \endverbatim
154: *>
155: *> \param[out] ITER
156: *> \verbatim
157: *> ITER is INTEGER
158: *> < 0: iterative refinement has failed, double precision
159: *> factorization has been performed
160: *> -1 : the routine fell back to full precision for
161: *> implementation- or machine-specific reasons
162: *> -2 : narrowing the precision induced an overflow,
163: *> the routine fell back to full precision
164: *> -3 : failure of SGETRF
165: *> -31: stop the iterative refinement after the 30th
166: *> iterations
167: *> > 0: iterative refinement has been sucessfully used.
168: *> Returns the number of iterations
169: *> \endverbatim
170: *>
171: *> \param[out] INFO
172: *> \verbatim
173: *> INFO is INTEGER
174: *> = 0: successful exit
175: *> < 0: if INFO = -i, the i-th argument had an illegal value
176: *> > 0: if INFO = i, U(i,i) computed in DOUBLE PRECISION is
177: *> exactly zero. The factorization has been completed,
178: *> but the factor U is exactly singular, so the solution
179: *> could not be computed.
180: *> \endverbatim
181: *
182: * Authors:
183: * ========
184: *
185: *> \author Univ. of Tennessee
186: *> \author Univ. of California Berkeley
187: *> \author Univ. of Colorado Denver
188: *> \author NAG Ltd.
189: *
190: *> \date November 2011
191: *
192: *> \ingroup doubleGEsolve
193: *
194: * =====================================================================
195: SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
196: $ SWORK, ITER, INFO )
197: *
198: * -- LAPACK driver routine (version 3.4.0) --
199: * -- LAPACK is a software package provided by Univ. of Tennessee, --
200: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
201: * November 2011
202: *
203: * .. Scalar Arguments ..
204: INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
205: * ..
206: * .. Array Arguments ..
207: INTEGER IPIV( * )
208: REAL SWORK( * )
209: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
210: $ X( LDX, * )
211: * ..
212: *
213: * =====================================================================
214: *
215: * .. Parameters ..
216: LOGICAL DOITREF
217: PARAMETER ( DOITREF = .TRUE. )
218: *
219: INTEGER ITERMAX
220: PARAMETER ( ITERMAX = 30 )
221: *
222: DOUBLE PRECISION BWDMAX
223: PARAMETER ( BWDMAX = 1.0E+00 )
224: *
225: DOUBLE PRECISION NEGONE, ONE
226: PARAMETER ( NEGONE = -1.0D+0, ONE = 1.0D+0 )
227: *
228: * .. Local Scalars ..
229: INTEGER I, IITER, PTSA, PTSX
230: DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
231: *
232: * .. External Subroutines ..
233: EXTERNAL DAXPY, DGEMM, DLACPY, DLAG2S, SLAG2D, SGETRF,
234: $ SGETRS, XERBLA
235: * ..
236: * .. External Functions ..
237: INTEGER IDAMAX
238: DOUBLE PRECISION DLAMCH, DLANGE
239: EXTERNAL IDAMAX, DLAMCH, DLANGE
240: * ..
241: * .. Intrinsic Functions ..
242: INTRINSIC ABS, DBLE, MAX, SQRT
243: * ..
244: * .. Executable Statements ..
245: *
246: INFO = 0
247: ITER = 0
248: *
249: * Test the input parameters.
250: *
251: IF( N.LT.0 ) THEN
252: INFO = -1
253: ELSE IF( NRHS.LT.0 ) THEN
254: INFO = -2
255: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
256: INFO = -4
257: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
258: INFO = -7
259: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
260: INFO = -9
261: END IF
262: IF( INFO.NE.0 ) THEN
263: CALL XERBLA( 'DSGESV', -INFO )
264: RETURN
265: END IF
266: *
267: * Quick return if (N.EQ.0).
268: *
269: IF( N.EQ.0 )
270: $ RETURN
271: *
272: * Skip single precision iterative refinement if a priori slower
273: * than double precision factorization.
274: *
275: IF( .NOT.DOITREF ) THEN
276: ITER = -1
277: GO TO 40
278: END IF
279: *
280: * Compute some constants.
281: *
282: ANRM = DLANGE( 'I', N, N, A, LDA, WORK )
283: EPS = DLAMCH( 'Epsilon' )
284: CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX
285: *
286: * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
287: *
288: PTSA = 1
289: PTSX = PTSA + N*N
290: *
291: * Convert B from double precision to single precision and store the
292: * result in SX.
293: *
294: CALL DLAG2S( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO )
295: *
296: IF( INFO.NE.0 ) THEN
297: ITER = -2
298: GO TO 40
299: END IF
300: *
301: * Convert A from double precision to single precision and store the
302: * result in SA.
303: *
304: CALL DLAG2S( N, N, A, LDA, SWORK( PTSA ), N, INFO )
305: *
306: IF( INFO.NE.0 ) THEN
307: ITER = -2
308: GO TO 40
309: END IF
310: *
311: * Compute the LU factorization of SA.
312: *
313: CALL SGETRF( N, N, SWORK( PTSA ), N, IPIV, INFO )
314: *
315: IF( INFO.NE.0 ) THEN
316: ITER = -3
317: GO TO 40
318: END IF
319: *
320: * Solve the system SA*SX = SB.
321: *
322: CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV,
323: $ SWORK( PTSX ), N, INFO )
324: *
325: * Convert SX back to double precision
326: *
327: CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO )
328: *
329: * Compute R = B - AX (R is WORK).
330: *
331: CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N )
332: *
333: CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, A,
334: $ LDA, X, LDX, ONE, WORK, N )
335: *
336: * Check whether the NRHS normwise backward errors satisfy the
337: * stopping criterion. If yes, set ITER=0 and return.
338: *
339: DO I = 1, NRHS
340: XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) )
341: RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) )
342: IF( RNRM.GT.XNRM*CTE )
343: $ GO TO 10
344: END DO
345: *
346: * If we are here, the NRHS normwise backward errors satisfy the
347: * stopping criterion. We are good to exit.
348: *
349: ITER = 0
350: RETURN
351: *
352: 10 CONTINUE
353: *
354: DO 30 IITER = 1, ITERMAX
355: *
356: * Convert R (in WORK) from double precision to single precision
357: * and store the result in SX.
358: *
359: CALL DLAG2S( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO )
360: *
361: IF( INFO.NE.0 ) THEN
362: ITER = -2
363: GO TO 40
364: END IF
365: *
366: * Solve the system SA*SX = SR.
367: *
368: CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV,
369: $ SWORK( PTSX ), N, INFO )
370: *
371: * Convert SX back to double precision and update the current
372: * iterate.
373: *
374: CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO )
375: *
376: DO I = 1, NRHS
377: CALL DAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 )
378: END DO
379: *
380: * Compute R = B - AX (R is WORK).
381: *
382: CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N )
383: *
384: CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE,
385: $ A, LDA, X, LDX, ONE, WORK, N )
386: *
387: * Check whether the NRHS normwise backward errors satisfy the
388: * stopping criterion. If yes, set ITER=IITER>0 and return.
389: *
390: DO I = 1, NRHS
391: XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) )
392: RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) )
393: IF( RNRM.GT.XNRM*CTE )
394: $ GO TO 20
395: END DO
396: *
397: * If we are here, the NRHS normwise backward errors satisfy the
398: * stopping criterion, we are good to exit.
399: *
400: ITER = IITER
401: *
402: RETURN
403: *
404: 20 CONTINUE
405: *
406: 30 CONTINUE
407: *
408: * If we are at this place of the code, this is because we have
409: * performed ITER=ITERMAX iterations and never satisified the
410: * stopping criterion, set up the ITER flag accordingly and follow up
411: * on double precision routine.
412: *
413: ITER = -ITERMAX - 1
414: *
415: 40 CONTINUE
416: *
417: * Single-precision iterative refinement failed to converge to a
418: * satisfactory solution, so we resort to double precision.
419: *
420: CALL DGETRF( N, N, A, LDA, IPIV, INFO )
421: *
422: IF( INFO.NE.0 )
423: $ RETURN
424: *
425: CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX )
426: CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX,
427: $ INFO )
428: *
429: RETURN
430: *
431: * End of DSGESV.
432: *
433: END
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