Annotation of rpl/lapack/lapack/dlasy2.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DLASY2
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLASY2 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasy2.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
! 22: * LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * LOGICAL LTRANL, LTRANR
! 26: * INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
! 27: * DOUBLE PRECISION SCALE, XNORM
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
! 31: * $ X( LDX, * )
! 32: * ..
! 33: *
! 34: *
! 35: *> \par Purpose:
! 36: * =============
! 37: *>
! 38: *> \verbatim
! 39: *>
! 40: *> DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
! 41: *>
! 42: *> op(TL)*X + ISGN*X*op(TR) = SCALE*B,
! 43: *>
! 44: *> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
! 45: *> -1. op(T) = T or T**T, where T**T denotes the transpose of T.
! 46: *> \endverbatim
! 47: *
! 48: * Arguments:
! 49: * ==========
! 50: *
! 51: *> \param[in] LTRANL
! 52: *> \verbatim
! 53: *> LTRANL is LOGICAL
! 54: *> On entry, LTRANL specifies the op(TL):
! 55: *> = .FALSE., op(TL) = TL,
! 56: *> = .TRUE., op(TL) = TL**T.
! 57: *> \endverbatim
! 58: *>
! 59: *> \param[in] LTRANR
! 60: *> \verbatim
! 61: *> LTRANR is LOGICAL
! 62: *> On entry, LTRANR specifies the op(TR):
! 63: *> = .FALSE., op(TR) = TR,
! 64: *> = .TRUE., op(TR) = TR**T.
! 65: *> \endverbatim
! 66: *>
! 67: *> \param[in] ISGN
! 68: *> \verbatim
! 69: *> ISGN is INTEGER
! 70: *> On entry, ISGN specifies the sign of the equation
! 71: *> as described before. ISGN may only be 1 or -1.
! 72: *> \endverbatim
! 73: *>
! 74: *> \param[in] N1
! 75: *> \verbatim
! 76: *> N1 is INTEGER
! 77: *> On entry, N1 specifies the order of matrix TL.
! 78: *> N1 may only be 0, 1 or 2.
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] N2
! 82: *> \verbatim
! 83: *> N2 is INTEGER
! 84: *> On entry, N2 specifies the order of matrix TR.
! 85: *> N2 may only be 0, 1 or 2.
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[in] TL
! 89: *> \verbatim
! 90: *> TL is DOUBLE PRECISION array, dimension (LDTL,2)
! 91: *> On entry, TL contains an N1 by N1 matrix.
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[in] LDTL
! 95: *> \verbatim
! 96: *> LDTL is INTEGER
! 97: *> The leading dimension of the matrix TL. LDTL >= max(1,N1).
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] TR
! 101: *> \verbatim
! 102: *> TR is DOUBLE PRECISION array, dimension (LDTR,2)
! 103: *> On entry, TR contains an N2 by N2 matrix.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] LDTR
! 107: *> \verbatim
! 108: *> LDTR is INTEGER
! 109: *> The leading dimension of the matrix TR. LDTR >= max(1,N2).
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] B
! 113: *> \verbatim
! 114: *> B is DOUBLE PRECISION array, dimension (LDB,2)
! 115: *> On entry, the N1 by N2 matrix B contains the right-hand
! 116: *> side of the equation.
! 117: *> \endverbatim
! 118: *>
! 119: *> \param[in] LDB
! 120: *> \verbatim
! 121: *> LDB is INTEGER
! 122: *> The leading dimension of the matrix B. LDB >= max(1,N1).
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[out] SCALE
! 126: *> \verbatim
! 127: *> SCALE is DOUBLE PRECISION
! 128: *> On exit, SCALE contains the scale factor. SCALE is chosen
! 129: *> less than or equal to 1 to prevent the solution overflowing.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[out] X
! 133: *> \verbatim
! 134: *> X is DOUBLE PRECISION array, dimension (LDX,2)
! 135: *> On exit, X contains the N1 by N2 solution.
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[in] LDX
! 139: *> \verbatim
! 140: *> LDX is INTEGER
! 141: *> The leading dimension of the matrix X. LDX >= max(1,N1).
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[out] XNORM
! 145: *> \verbatim
! 146: *> XNORM is DOUBLE PRECISION
! 147: *> On exit, XNORM is the infinity-norm of the solution.
! 148: *> \endverbatim
! 149: *>
! 150: *> \param[out] INFO
! 151: *> \verbatim
! 152: *> INFO is INTEGER
! 153: *> On exit, INFO is set to
! 154: *> 0: successful exit.
! 155: *> 1: TL and TR have too close eigenvalues, so TL or
! 156: *> TR is perturbed to get a nonsingular equation.
! 157: *> NOTE: In the interests of speed, this routine does not
! 158: *> check the inputs for errors.
! 159: *> \endverbatim
! 160: *
! 161: * Authors:
! 162: * ========
! 163: *
! 164: *> \author Univ. of Tennessee
! 165: *> \author Univ. of California Berkeley
! 166: *> \author Univ. of Colorado Denver
! 167: *> \author NAG Ltd.
! 168: *
! 169: *> \date November 2011
! 170: *
! 171: *> \ingroup doubleSYauxiliary
! 172: *
! 173: * =====================================================================
1.1 bertrand 174: SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
175: $ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
176: *
1.9 ! bertrand 177: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 178: * -- LAPACK is a software package provided by Univ. of Tennessee, --
179: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 180: * November 2011
1.1 bertrand 181: *
182: * .. Scalar Arguments ..
183: LOGICAL LTRANL, LTRANR
184: INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
185: DOUBLE PRECISION SCALE, XNORM
186: * ..
187: * .. Array Arguments ..
188: DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
189: $ X( LDX, * )
190: * ..
191: *
192: * =====================================================================
193: *
194: * .. Parameters ..
195: DOUBLE PRECISION ZERO, ONE
196: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
197: DOUBLE PRECISION TWO, HALF, EIGHT
198: PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
199: * ..
200: * .. Local Scalars ..
201: LOGICAL BSWAP, XSWAP
202: INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
203: DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
204: $ TEMP, U11, U12, U22, XMAX
205: * ..
206: * .. Local Arrays ..
207: LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
208: INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
209: $ LOCU22( 4 )
210: DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
211: * ..
212: * .. External Functions ..
213: INTEGER IDAMAX
214: DOUBLE PRECISION DLAMCH
215: EXTERNAL IDAMAX, DLAMCH
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL DCOPY, DSWAP
219: * ..
220: * .. Intrinsic Functions ..
221: INTRINSIC ABS, MAX
222: * ..
223: * .. Data statements ..
224: DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
225: $ LOCU22 / 4, 3, 2, 1 /
226: DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
227: DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
228: * ..
229: * .. Executable Statements ..
230: *
231: * Do not check the input parameters for errors
232: *
233: INFO = 0
234: *
235: * Quick return if possible
236: *
237: IF( N1.EQ.0 .OR. N2.EQ.0 )
238: $ RETURN
239: *
240: * Set constants to control overflow
241: *
242: EPS = DLAMCH( 'P' )
243: SMLNUM = DLAMCH( 'S' ) / EPS
244: SGN = ISGN
245: *
246: K = N1 + N1 + N2 - 2
247: GO TO ( 10, 20, 30, 50 )K
248: *
249: * 1 by 1: TL11*X + SGN*X*TR11 = B11
250: *
251: 10 CONTINUE
252: TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
253: BET = ABS( TAU1 )
254: IF( BET.LE.SMLNUM ) THEN
255: TAU1 = SMLNUM
256: BET = SMLNUM
257: INFO = 1
258: END IF
259: *
260: SCALE = ONE
261: GAM = ABS( B( 1, 1 ) )
262: IF( SMLNUM*GAM.GT.BET )
263: $ SCALE = ONE / GAM
264: *
265: X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
266: XNORM = ABS( X( 1, 1 ) )
267: RETURN
268: *
269: * 1 by 2:
270: * TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
271: * [TR21 TR22]
272: *
273: 20 CONTINUE
274: *
275: SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
276: $ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
277: $ SMLNUM )
278: TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
279: TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
280: IF( LTRANR ) THEN
281: TMP( 2 ) = SGN*TR( 2, 1 )
282: TMP( 3 ) = SGN*TR( 1, 2 )
283: ELSE
284: TMP( 2 ) = SGN*TR( 1, 2 )
285: TMP( 3 ) = SGN*TR( 2, 1 )
286: END IF
287: BTMP( 1 ) = B( 1, 1 )
288: BTMP( 2 ) = B( 1, 2 )
289: GO TO 40
290: *
291: * 2 by 1:
292: * op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
293: * [TL21 TL22] [X21] [X21] [B21]
294: *
295: 30 CONTINUE
296: SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
297: $ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
298: $ SMLNUM )
299: TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
300: TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
301: IF( LTRANL ) THEN
302: TMP( 2 ) = TL( 1, 2 )
303: TMP( 3 ) = TL( 2, 1 )
304: ELSE
305: TMP( 2 ) = TL( 2, 1 )
306: TMP( 3 ) = TL( 1, 2 )
307: END IF
308: BTMP( 1 ) = B( 1, 1 )
309: BTMP( 2 ) = B( 2, 1 )
310: 40 CONTINUE
311: *
312: * Solve 2 by 2 system using complete pivoting.
313: * Set pivots less than SMIN to SMIN.
314: *
315: IPIV = IDAMAX( 4, TMP, 1 )
316: U11 = TMP( IPIV )
317: IF( ABS( U11 ).LE.SMIN ) THEN
318: INFO = 1
319: U11 = SMIN
320: END IF
321: U12 = TMP( LOCU12( IPIV ) )
322: L21 = TMP( LOCL21( IPIV ) ) / U11
323: U22 = TMP( LOCU22( IPIV ) ) - U12*L21
324: XSWAP = XSWPIV( IPIV )
325: BSWAP = BSWPIV( IPIV )
326: IF( ABS( U22 ).LE.SMIN ) THEN
327: INFO = 1
328: U22 = SMIN
329: END IF
330: IF( BSWAP ) THEN
331: TEMP = BTMP( 2 )
332: BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
333: BTMP( 1 ) = TEMP
334: ELSE
335: BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
336: END IF
337: SCALE = ONE
338: IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
339: $ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
340: SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
341: BTMP( 1 ) = BTMP( 1 )*SCALE
342: BTMP( 2 ) = BTMP( 2 )*SCALE
343: END IF
344: X2( 2 ) = BTMP( 2 ) / U22
345: X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
346: IF( XSWAP ) THEN
347: TEMP = X2( 2 )
348: X2( 2 ) = X2( 1 )
349: X2( 1 ) = TEMP
350: END IF
351: X( 1, 1 ) = X2( 1 )
352: IF( N1.EQ.1 ) THEN
353: X( 1, 2 ) = X2( 2 )
354: XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
355: ELSE
356: X( 2, 1 ) = X2( 2 )
357: XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
358: END IF
359: RETURN
360: *
361: * 2 by 2:
362: * op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
363: * [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
364: *
365: * Solve equivalent 4 by 4 system using complete pivoting.
366: * Set pivots less than SMIN to SMIN.
367: *
368: 50 CONTINUE
369: SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
370: $ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
371: SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
372: $ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
373: SMIN = MAX( EPS*SMIN, SMLNUM )
374: BTMP( 1 ) = ZERO
375: CALL DCOPY( 16, BTMP, 0, T16, 1 )
376: T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
377: T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
378: T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
379: T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
380: IF( LTRANL ) THEN
381: T16( 1, 2 ) = TL( 2, 1 )
382: T16( 2, 1 ) = TL( 1, 2 )
383: T16( 3, 4 ) = TL( 2, 1 )
384: T16( 4, 3 ) = TL( 1, 2 )
385: ELSE
386: T16( 1, 2 ) = TL( 1, 2 )
387: T16( 2, 1 ) = TL( 2, 1 )
388: T16( 3, 4 ) = TL( 1, 2 )
389: T16( 4, 3 ) = TL( 2, 1 )
390: END IF
391: IF( LTRANR ) THEN
392: T16( 1, 3 ) = SGN*TR( 1, 2 )
393: T16( 2, 4 ) = SGN*TR( 1, 2 )
394: T16( 3, 1 ) = SGN*TR( 2, 1 )
395: T16( 4, 2 ) = SGN*TR( 2, 1 )
396: ELSE
397: T16( 1, 3 ) = SGN*TR( 2, 1 )
398: T16( 2, 4 ) = SGN*TR( 2, 1 )
399: T16( 3, 1 ) = SGN*TR( 1, 2 )
400: T16( 4, 2 ) = SGN*TR( 1, 2 )
401: END IF
402: BTMP( 1 ) = B( 1, 1 )
403: BTMP( 2 ) = B( 2, 1 )
404: BTMP( 3 ) = B( 1, 2 )
405: BTMP( 4 ) = B( 2, 2 )
406: *
407: * Perform elimination
408: *
409: DO 100 I = 1, 3
410: XMAX = ZERO
411: DO 70 IP = I, 4
412: DO 60 JP = I, 4
413: IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
414: XMAX = ABS( T16( IP, JP ) )
415: IPSV = IP
416: JPSV = JP
417: END IF
418: 60 CONTINUE
419: 70 CONTINUE
420: IF( IPSV.NE.I ) THEN
421: CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
422: TEMP = BTMP( I )
423: BTMP( I ) = BTMP( IPSV )
424: BTMP( IPSV ) = TEMP
425: END IF
426: IF( JPSV.NE.I )
427: $ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
428: JPIV( I ) = JPSV
429: IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
430: INFO = 1
431: T16( I, I ) = SMIN
432: END IF
433: DO 90 J = I + 1, 4
434: T16( J, I ) = T16( J, I ) / T16( I, I )
435: BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
436: DO 80 K = I + 1, 4
437: T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
438: 80 CONTINUE
439: 90 CONTINUE
440: 100 CONTINUE
441: IF( ABS( T16( 4, 4 ) ).LT.SMIN )
442: $ T16( 4, 4 ) = SMIN
443: SCALE = ONE
444: IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
445: $ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
446: $ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
447: $ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
448: SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
449: $ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
450: BTMP( 1 ) = BTMP( 1 )*SCALE
451: BTMP( 2 ) = BTMP( 2 )*SCALE
452: BTMP( 3 ) = BTMP( 3 )*SCALE
453: BTMP( 4 ) = BTMP( 4 )*SCALE
454: END IF
455: DO 120 I = 1, 4
456: K = 5 - I
457: TEMP = ONE / T16( K, K )
458: TMP( K ) = BTMP( K )*TEMP
459: DO 110 J = K + 1, 4
460: TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
461: 110 CONTINUE
462: 120 CONTINUE
463: DO 130 I = 1, 3
464: IF( JPIV( 4-I ).NE.4-I ) THEN
465: TEMP = TMP( 4-I )
466: TMP( 4-I ) = TMP( JPIV( 4-I ) )
467: TMP( JPIV( 4-I ) ) = TEMP
468: END IF
469: 130 CONTINUE
470: X( 1, 1 ) = TMP( 1 )
471: X( 2, 1 ) = TMP( 2 )
472: X( 1, 2 ) = TMP( 3 )
473: X( 2, 2 ) = TMP( 4 )
474: XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
475: $ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
476: RETURN
477: *
478: * End of DLASY2
479: *
480: END
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