Annotation of rpl/lapack/lapack/zpftri.f, revision 1.7
1.7 ! bertrand 1: *> \brief \b ZPFTRI
! 2: *
! 3: * =========== DOCUMENTATION ===========
1.1 bertrand 4: *
1.7 ! bertrand 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
1.1 bertrand 7: *
1.7 ! bertrand 8: *> \htmlonly
! 9: *> Download ZPFTRI + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpftri.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpftri.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpftri.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER TRANSR, UPLO
! 25: * INTEGER INFO, N
! 26: * .. Array Arguments ..
! 27: * COMPLEX*16 A( 0: * )
! 28: * ..
! 29: *
! 30: *
! 31: *> \par Purpose:
! 32: * =============
! 33: *>
! 34: *> \verbatim
! 35: *>
! 36: *> ZPFTRI computes the inverse of a complex Hermitian positive definite
! 37: *> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
! 38: *> computed by ZPFTRF.
! 39: *> \endverbatim
! 40: *
! 41: * Arguments:
! 42: * ==========
! 43: *
! 44: *> \param[in] TRANSR
! 45: *> \verbatim
! 46: *> TRANSR is CHARACTER*1
! 47: *> = 'N': The Normal TRANSR of RFP A is stored;
! 48: *> = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
! 49: *> \endverbatim
! 50: *>
! 51: *> \param[in] UPLO
! 52: *> \verbatim
! 53: *> UPLO is CHARACTER*1
! 54: *> = 'U': Upper triangle of A is stored;
! 55: *> = 'L': Lower triangle of A is stored.
! 56: *> \endverbatim
! 57: *>
! 58: *> \param[in] N
! 59: *> \verbatim
! 60: *> N is INTEGER
! 61: *> The order of the matrix A. N >= 0.
! 62: *> \endverbatim
! 63: *>
! 64: *> \param[in,out] A
! 65: *> \verbatim
! 66: *> A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
! 67: *> On entry, the Hermitian matrix A in RFP format. RFP format is
! 68: *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
! 69: *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
! 70: *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
! 71: *> the Conjugate-transpose of RFP A as defined when
! 72: *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
! 73: *> follows: If UPLO = 'U' the RFP A contains the nt elements of
! 74: *> upper packed A. If UPLO = 'L' the RFP A contains the elements
! 75: *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
! 76: *> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
! 77: *> is odd. See the Note below for more details.
! 78: *>
! 79: *> On exit, the Hermitian inverse of the original matrix, in the
! 80: *> same storage format.
! 81: *> \endverbatim
! 82: *>
! 83: *> \param[out] INFO
! 84: *> \verbatim
! 85: *> INFO is INTEGER
! 86: *> = 0: successful exit
! 87: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 88: *> > 0: if INFO = i, the (i,i) element of the factor U or L is
! 89: *> zero, and the inverse could not be computed.
! 90: *> \endverbatim
! 91: *
! 92: * Authors:
! 93: * ========
! 94: *
! 95: *> \author Univ. of Tennessee
! 96: *> \author Univ. of California Berkeley
! 97: *> \author Univ. of Colorado Denver
! 98: *> \author NAG Ltd.
! 99: *
! 100: *> \date November 2011
! 101: *
! 102: *> \ingroup complex16OTHERcomputational
! 103: *
! 104: *> \par Further Details:
! 105: * =====================
! 106: *>
! 107: *> \verbatim
! 108: *>
! 109: *> We first consider Standard Packed Format when N is even.
! 110: *> We give an example where N = 6.
! 111: *>
! 112: *> AP is Upper AP is Lower
! 113: *>
! 114: *> 00 01 02 03 04 05 00
! 115: *> 11 12 13 14 15 10 11
! 116: *> 22 23 24 25 20 21 22
! 117: *> 33 34 35 30 31 32 33
! 118: *> 44 45 40 41 42 43 44
! 119: *> 55 50 51 52 53 54 55
! 120: *>
! 121: *>
! 122: *> Let TRANSR = 'N'. RFP holds AP as follows:
! 123: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
! 124: *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
! 125: *> conjugate-transpose of the first three columns of AP upper.
! 126: *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
! 127: *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
! 128: *> conjugate-transpose of the last three columns of AP lower.
! 129: *> To denote conjugate we place -- above the element. This covers the
! 130: *> case N even and TRANSR = 'N'.
! 131: *>
! 132: *> RFP A RFP A
! 133: *>
! 134: *> -- -- --
! 135: *> 03 04 05 33 43 53
! 136: *> -- --
! 137: *> 13 14 15 00 44 54
! 138: *> --
! 139: *> 23 24 25 10 11 55
! 140: *>
! 141: *> 33 34 35 20 21 22
! 142: *> --
! 143: *> 00 44 45 30 31 32
! 144: *> -- --
! 145: *> 01 11 55 40 41 42
! 146: *> -- -- --
! 147: *> 02 12 22 50 51 52
! 148: *>
! 149: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
! 150: *> transpose of RFP A above. One therefore gets:
! 151: *>
! 152: *>
! 153: *> RFP A RFP A
! 154: *>
! 155: *> -- -- -- -- -- -- -- -- -- --
! 156: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
! 157: *> -- -- -- -- -- -- -- -- -- --
! 158: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
! 159: *> -- -- -- -- -- -- -- -- -- --
! 160: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
! 161: *>
! 162: *>
! 163: *> We next consider Standard Packed Format when N is odd.
! 164: *> We give an example where N = 5.
! 165: *>
! 166: *> AP is Upper AP is Lower
! 167: *>
! 168: *> 00 01 02 03 04 00
! 169: *> 11 12 13 14 10 11
! 170: *> 22 23 24 20 21 22
! 171: *> 33 34 30 31 32 33
! 172: *> 44 40 41 42 43 44
! 173: *>
! 174: *>
! 175: *> Let TRANSR = 'N'. RFP holds AP as follows:
! 176: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
! 177: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
! 178: *> conjugate-transpose of the first two columns of AP upper.
! 179: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
! 180: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
! 181: *> conjugate-transpose of the last two columns of AP lower.
! 182: *> To denote conjugate we place -- above the element. This covers the
! 183: *> case N odd and TRANSR = 'N'.
! 184: *>
! 185: *> RFP A RFP A
! 186: *>
! 187: *> -- --
! 188: *> 02 03 04 00 33 43
! 189: *> --
! 190: *> 12 13 14 10 11 44
! 191: *>
! 192: *> 22 23 24 20 21 22
! 193: *> --
! 194: *> 00 33 34 30 31 32
! 195: *> -- --
! 196: *> 01 11 44 40 41 42
! 197: *>
! 198: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
! 199: *> transpose of RFP A above. One therefore gets:
! 200: *>
! 201: *>
! 202: *> RFP A RFP A
! 203: *>
! 204: *> -- -- -- -- -- -- -- -- --
! 205: *> 02 12 22 00 01 00 10 20 30 40 50
! 206: *> -- -- -- -- -- -- -- -- --
! 207: *> 03 13 23 33 11 33 11 21 31 41 51
! 208: *> -- -- -- -- -- -- -- -- --
! 209: *> 04 14 24 34 44 43 44 22 32 42 52
! 210: *> \endverbatim
! 211: *>
! 212: * =====================================================================
! 213: SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
1.1 bertrand 214: *
1.7 ! bertrand 215: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 216: * -- LAPACK is a software package provided by Univ. of Tennessee, --
217: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.7 ! bertrand 218: * November 2011
1.1 bertrand 219: *
220: * .. Scalar Arguments ..
221: CHARACTER TRANSR, UPLO
222: INTEGER INFO, N
223: * .. Array Arguments ..
224: COMPLEX*16 A( 0: * )
225: * ..
226: *
227: * =====================================================================
228: *
229: * .. Parameters ..
230: DOUBLE PRECISION ONE
231: COMPLEX*16 CONE
232: PARAMETER ( ONE = 1.D0, CONE = ( 1.D0, 0.D0 ) )
233: * ..
234: * .. Local Scalars ..
235: LOGICAL LOWER, NISODD, NORMALTRANSR
236: INTEGER N1, N2, K
237: * ..
238: * .. External Functions ..
239: LOGICAL LSAME
240: EXTERNAL LSAME
241: * ..
242: * .. External Subroutines ..
243: EXTERNAL XERBLA, ZTFTRI, ZLAUUM, ZTRMM, ZHERK
244: * ..
245: * .. Intrinsic Functions ..
246: INTRINSIC MOD
247: * ..
248: * .. Executable Statements ..
249: *
250: * Test the input parameters.
251: *
252: INFO = 0
253: NORMALTRANSR = LSAME( TRANSR, 'N' )
254: LOWER = LSAME( UPLO, 'L' )
255: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
256: INFO = -1
257: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
258: INFO = -2
259: ELSE IF( N.LT.0 ) THEN
260: INFO = -3
261: END IF
262: IF( INFO.NE.0 ) THEN
263: CALL XERBLA( 'ZPFTRI', -INFO )
264: RETURN
265: END IF
266: *
267: * Quick return if possible
268: *
269: IF( N.EQ.0 )
1.6 bertrand 270: $ RETURN
1.1 bertrand 271: *
272: * Invert the triangular Cholesky factor U or L.
273: *
274: CALL ZTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
275: IF( INFO.GT.0 )
1.6 bertrand 276: $ RETURN
1.1 bertrand 277: *
278: * If N is odd, set NISODD = .TRUE.
279: * If N is even, set K = N/2 and NISODD = .FALSE.
280: *
281: IF( MOD( N, 2 ).EQ.0 ) THEN
282: K = N / 2
283: NISODD = .FALSE.
284: ELSE
285: NISODD = .TRUE.
286: END IF
287: *
288: * Set N1 and N2 depending on LOWER
289: *
290: IF( LOWER ) THEN
291: N2 = N / 2
292: N1 = N - N2
293: ELSE
294: N1 = N / 2
295: N2 = N - N1
296: END IF
297: *
298: * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
299: * inv(L)^C*inv(L). There are eight cases.
300: *
301: IF( NISODD ) THEN
302: *
303: * N is odd
304: *
305: IF( NORMALTRANSR ) THEN
306: *
307: * N is odd and TRANSR = 'N'
308: *
309: IF( LOWER ) THEN
310: *
311: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
312: * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
313: * T1 -> a(0), T2 -> a(n), S -> a(N1)
314: *
315: CALL ZLAUUM( 'L', N1, A( 0 ), N, INFO )
316: CALL ZHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE,
1.6 bertrand 317: $ A( 0 ), N )
1.1 bertrand 318: CALL ZTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N,
1.6 bertrand 319: $ A( N1 ), N )
1.1 bertrand 320: CALL ZLAUUM( 'U', N2, A( N ), N, INFO )
321: *
322: ELSE
323: *
324: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
325: * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
326: * T1 -> a(N2), T2 -> a(N1), S -> a(0)
327: *
328: CALL ZLAUUM( 'L', N1, A( N2 ), N, INFO )
329: CALL ZHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
1.6 bertrand 330: $ A( N2 ), N )
1.1 bertrand 331: CALL ZTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N,
1.6 bertrand 332: $ A( 0 ), N )
1.1 bertrand 333: CALL ZLAUUM( 'U', N2, A( N1 ), N, INFO )
334: *
335: END IF
336: *
337: ELSE
338: *
339: * N is odd and TRANSR = 'C'
340: *
341: IF( LOWER ) THEN
342: *
343: * SRPA for LOWER, TRANSPOSE, and N is odd
344: * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
345: *
346: CALL ZLAUUM( 'U', N1, A( 0 ), N1, INFO )
347: CALL ZHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
1.6 bertrand 348: $ A( 0 ), N1 )
1.1 bertrand 349: CALL ZTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1,
1.6 bertrand 350: $ A( N1*N1 ), N1 )
1.1 bertrand 351: CALL ZLAUUM( 'L', N2, A( 1 ), N1, INFO )
352: *
353: ELSE
354: *
355: * SRPA for UPPER, TRANSPOSE, and N is odd
356: * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
357: *
358: CALL ZLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
359: CALL ZHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE,
1.6 bertrand 360: $ A( N2*N2 ), N2 )
1.1 bertrand 361: CALL ZTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ),
1.6 bertrand 362: $ N2, A( 0 ), N2 )
1.1 bertrand 363: CALL ZLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
364: *
365: END IF
366: *
367: END IF
368: *
369: ELSE
370: *
371: * N is even
372: *
373: IF( NORMALTRANSR ) THEN
374: *
375: * N is even and TRANSR = 'N'
376: *
377: IF( LOWER ) THEN
378: *
379: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
380: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
381: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
382: *
383: CALL ZLAUUM( 'L', K, A( 1 ), N+1, INFO )
384: CALL ZHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE,
1.6 bertrand 385: $ A( 1 ), N+1 )
1.1 bertrand 386: CALL ZTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1,
1.6 bertrand 387: $ A( K+1 ), N+1 )
1.1 bertrand 388: CALL ZLAUUM( 'U', K, A( 0 ), N+1, INFO )
389: *
390: ELSE
391: *
392: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
393: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
394: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
395: *
396: CALL ZLAUUM( 'L', K, A( K+1 ), N+1, INFO )
397: CALL ZHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
1.6 bertrand 398: $ A( K+1 ), N+1 )
1.1 bertrand 399: CALL ZTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1,
1.6 bertrand 400: $ A( 0 ), N+1 )
1.1 bertrand 401: CALL ZLAUUM( 'U', K, A( K ), N+1, INFO )
402: *
403: END IF
404: *
405: ELSE
406: *
407: * N is even and TRANSR = 'C'
408: *
409: IF( LOWER ) THEN
410: *
411: * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
412: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
413: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
414: *
415: CALL ZLAUUM( 'U', K, A( K ), K, INFO )
416: CALL ZHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
1.6 bertrand 417: $ A( K ), K )
1.1 bertrand 418: CALL ZTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K,
1.6 bertrand 419: $ A( K*( K+1 ) ), K )
1.1 bertrand 420: CALL ZLAUUM( 'L', K, A( 0 ), K, INFO )
421: *
422: ELSE
423: *
424: * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
425: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
426: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
427: *
428: CALL ZLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
429: CALL ZHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE,
1.6 bertrand 430: $ A( K*( K+1 ) ), K )
1.1 bertrand 431: CALL ZTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K,
1.6 bertrand 432: $ A( 0 ), K )
1.1 bertrand 433: CALL ZLAUUM( 'L', K, A( K*K ), K, INFO )
434: *
435: END IF
436: *
437: END IF
438: *
439: END IF
440: *
441: RETURN
442: *
443: * End of ZPFTRI
444: *
445: END
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