Annotation of rpl/lapack/lapack/zpftri.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2.1) --
! 4: *
! 5: * -- Contributed by Fred Gustavson of the IBM Watson Research Center --
! 6: * -- April 2009 --
! 7: *
! 8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 9: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER TRANSR, UPLO
! 13: INTEGER INFO, N
! 14: * .. Array Arguments ..
! 15: COMPLEX*16 A( 0: * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * ZPFTRI computes the inverse of a complex Hermitian positive definite
! 22: * matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
! 23: * computed by ZPFTRF.
! 24: *
! 25: * Arguments
! 26: * =========
! 27: *
! 28: * TRANSR (input) CHARACTER
! 29: * = 'N': The Normal TRANSR of RFP A is stored;
! 30: * = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
! 31: *
! 32: * UPLO (input) CHARACTER
! 33: * = 'U': Upper triangle of A is stored;
! 34: * = 'L': Lower triangle of A is stored.
! 35: *
! 36: * N (input) INTEGER
! 37: * The order of the matrix A. N >= 0.
! 38: *
! 39: * A (input/output) COMPLEX*16 array, dimension ( N*(N+1)/2 );
! 40: * On entry, the Hermitian matrix A in RFP format. RFP format is
! 41: * described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
! 42: * then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
! 43: * (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
! 44: * the Conjugate-transpose of RFP A as defined when
! 45: * TRANSR = 'N'. The contents of RFP A are defined by UPLO as
! 46: * follows: If UPLO = 'U' the RFP A contains the nt elements of
! 47: * upper packed A. If UPLO = 'L' the RFP A contains the elements
! 48: * of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
! 49: * 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
! 50: * is odd. See the Note below for more details.
! 51: *
! 52: * On exit, the Hermitian inverse of the original matrix, in the
! 53: * same storage format.
! 54: *
! 55: * INFO (output) INTEGER
! 56: * = 0: successful exit
! 57: * < 0: if INFO = -i, the i-th argument had an illegal value
! 58: * > 0: if INFO = i, the (i,i) element of the factor U or L is
! 59: * zero, and the inverse could not be computed.
! 60: *
! 61: * Further Details
! 62: * ===============
! 63: *
! 64: * We first consider Standard Packed Format when N is even.
! 65: * We give an example where N = 6.
! 66: *
! 67: * AP is Upper AP is Lower
! 68: *
! 69: * 00 01 02 03 04 05 00
! 70: * 11 12 13 14 15 10 11
! 71: * 22 23 24 25 20 21 22
! 72: * 33 34 35 30 31 32 33
! 73: * 44 45 40 41 42 43 44
! 74: * 55 50 51 52 53 54 55
! 75: *
! 76: *
! 77: * Let TRANSR = 'N'. RFP holds AP as follows:
! 78: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
! 79: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
! 80: * conjugate-transpose of the first three columns of AP upper.
! 81: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
! 82: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
! 83: * conjugate-transpose of the last three columns of AP lower.
! 84: * To denote conjugate we place -- above the element. This covers the
! 85: * case N even and TRANSR = 'N'.
! 86: *
! 87: * RFP A RFP A
! 88: *
! 89: * -- -- --
! 90: * 03 04 05 33 43 53
! 91: * -- --
! 92: * 13 14 15 00 44 54
! 93: * --
! 94: * 23 24 25 10 11 55
! 95: *
! 96: * 33 34 35 20 21 22
! 97: * --
! 98: * 00 44 45 30 31 32
! 99: * -- --
! 100: * 01 11 55 40 41 42
! 101: * -- -- --
! 102: * 02 12 22 50 51 52
! 103: *
! 104: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
! 105: * transpose of RFP A above. One therefore gets:
! 106: *
! 107: *
! 108: * RFP A RFP A
! 109: *
! 110: * -- -- -- -- -- -- -- -- -- --
! 111: * 03 13 23 33 00 01 02 33 00 10 20 30 40 50
! 112: * -- -- -- -- -- -- -- -- -- --
! 113: * 04 14 24 34 44 11 12 43 44 11 21 31 41 51
! 114: * -- -- -- -- -- -- -- -- -- --
! 115: * 05 15 25 35 45 55 22 53 54 55 22 32 42 52
! 116: *
! 117: *
! 118: * We next consider Standard Packed Format when N is odd.
! 119: * We give an example where N = 5.
! 120: *
! 121: * AP is Upper AP is Lower
! 122: *
! 123: * 00 01 02 03 04 00
! 124: * 11 12 13 14 10 11
! 125: * 22 23 24 20 21 22
! 126: * 33 34 30 31 32 33
! 127: * 44 40 41 42 43 44
! 128: *
! 129: *
! 130: * Let TRANSR = 'N'. RFP holds AP as follows:
! 131: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
! 132: * three columns of AP upper. The lower triangle A(3:4,0:1) consists of
! 133: * conjugate-transpose of the first two columns of AP upper.
! 134: * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
! 135: * three columns of AP lower. The upper triangle A(0:1,1:2) consists of
! 136: * conjugate-transpose of the last two columns of AP lower.
! 137: * To denote conjugate we place -- above the element. This covers the
! 138: * case N odd and TRANSR = 'N'.
! 139: *
! 140: * RFP A RFP A
! 141: *
! 142: * -- --
! 143: * 02 03 04 00 33 43
! 144: * --
! 145: * 12 13 14 10 11 44
! 146: *
! 147: * 22 23 24 20 21 22
! 148: * --
! 149: * 00 33 34 30 31 32
! 150: * -- --
! 151: * 01 11 44 40 41 42
! 152: *
! 153: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
! 154: * transpose of RFP A above. One therefore gets:
! 155: *
! 156: *
! 157: * RFP A RFP A
! 158: *
! 159: * -- -- -- -- -- -- -- -- --
! 160: * 02 12 22 00 01 00 10 20 30 40 50
! 161: * -- -- -- -- -- -- -- -- --
! 162: * 03 13 23 33 11 33 11 21 31 41 51
! 163: * -- -- -- -- -- -- -- -- --
! 164: * 04 14 24 34 44 43 44 22 32 42 52
! 165: *
! 166: * =====================================================================
! 167: *
! 168: * .. Parameters ..
! 169: DOUBLE PRECISION ONE
! 170: COMPLEX*16 CONE
! 171: PARAMETER ( ONE = 1.D0, CONE = ( 1.D0, 0.D0 ) )
! 172: * ..
! 173: * .. Local Scalars ..
! 174: LOGICAL LOWER, NISODD, NORMALTRANSR
! 175: INTEGER N1, N2, K
! 176: * ..
! 177: * .. External Functions ..
! 178: LOGICAL LSAME
! 179: EXTERNAL LSAME
! 180: * ..
! 181: * .. External Subroutines ..
! 182: EXTERNAL XERBLA, ZTFTRI, ZLAUUM, ZTRMM, ZHERK
! 183: * ..
! 184: * .. Intrinsic Functions ..
! 185: INTRINSIC MOD
! 186: * ..
! 187: * .. Executable Statements ..
! 188: *
! 189: * Test the input parameters.
! 190: *
! 191: INFO = 0
! 192: NORMALTRANSR = LSAME( TRANSR, 'N' )
! 193: LOWER = LSAME( UPLO, 'L' )
! 194: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
! 195: INFO = -1
! 196: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
! 197: INFO = -2
! 198: ELSE IF( N.LT.0 ) THEN
! 199: INFO = -3
! 200: END IF
! 201: IF( INFO.NE.0 ) THEN
! 202: CALL XERBLA( 'ZPFTRI', -INFO )
! 203: RETURN
! 204: END IF
! 205: *
! 206: * Quick return if possible
! 207: *
! 208: IF( N.EQ.0 )
! 209: + RETURN
! 210: *
! 211: * Invert the triangular Cholesky factor U or L.
! 212: *
! 213: CALL ZTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
! 214: IF( INFO.GT.0 )
! 215: + RETURN
! 216: *
! 217: * If N is odd, set NISODD = .TRUE.
! 218: * If N is even, set K = N/2 and NISODD = .FALSE.
! 219: *
! 220: IF( MOD( N, 2 ).EQ.0 ) THEN
! 221: K = N / 2
! 222: NISODD = .FALSE.
! 223: ELSE
! 224: NISODD = .TRUE.
! 225: END IF
! 226: *
! 227: * Set N1 and N2 depending on LOWER
! 228: *
! 229: IF( LOWER ) THEN
! 230: N2 = N / 2
! 231: N1 = N - N2
! 232: ELSE
! 233: N1 = N / 2
! 234: N2 = N - N1
! 235: END IF
! 236: *
! 237: * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
! 238: * inv(L)^C*inv(L). There are eight cases.
! 239: *
! 240: IF( NISODD ) THEN
! 241: *
! 242: * N is odd
! 243: *
! 244: IF( NORMALTRANSR ) THEN
! 245: *
! 246: * N is odd and TRANSR = 'N'
! 247: *
! 248: IF( LOWER ) THEN
! 249: *
! 250: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
! 251: * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
! 252: * T1 -> a(0), T2 -> a(n), S -> a(N1)
! 253: *
! 254: CALL ZLAUUM( 'L', N1, A( 0 ), N, INFO )
! 255: CALL ZHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE,
! 256: + A( 0 ), N )
! 257: CALL ZTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N,
! 258: + A( N1 ), N )
! 259: CALL ZLAUUM( 'U', N2, A( N ), N, INFO )
! 260: *
! 261: ELSE
! 262: *
! 263: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
! 264: * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
! 265: * T1 -> a(N2), T2 -> a(N1), S -> a(0)
! 266: *
! 267: CALL ZLAUUM( 'L', N1, A( N2 ), N, INFO )
! 268: CALL ZHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
! 269: + A( N2 ), N )
! 270: CALL ZTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N,
! 271: + A( 0 ), N )
! 272: CALL ZLAUUM( 'U', N2, A( N1 ), N, INFO )
! 273: *
! 274: END IF
! 275: *
! 276: ELSE
! 277: *
! 278: * N is odd and TRANSR = 'C'
! 279: *
! 280: IF( LOWER ) THEN
! 281: *
! 282: * SRPA for LOWER, TRANSPOSE, and N is odd
! 283: * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
! 284: *
! 285: CALL ZLAUUM( 'U', N1, A( 0 ), N1, INFO )
! 286: CALL ZHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
! 287: + A( 0 ), N1 )
! 288: CALL ZTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1,
! 289: + A( N1*N1 ), N1 )
! 290: CALL ZLAUUM( 'L', N2, A( 1 ), N1, INFO )
! 291: *
! 292: ELSE
! 293: *
! 294: * SRPA for UPPER, TRANSPOSE, and N is odd
! 295: * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
! 296: *
! 297: CALL ZLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
! 298: CALL ZHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE,
! 299: + A( N2*N2 ), N2 )
! 300: CALL ZTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ),
! 301: + N2, A( 0 ), N2 )
! 302: CALL ZLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
! 303: *
! 304: END IF
! 305: *
! 306: END IF
! 307: *
! 308: ELSE
! 309: *
! 310: * N is even
! 311: *
! 312: IF( NORMALTRANSR ) THEN
! 313: *
! 314: * N is even and TRANSR = 'N'
! 315: *
! 316: IF( LOWER ) THEN
! 317: *
! 318: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
! 319: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
! 320: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
! 321: *
! 322: CALL ZLAUUM( 'L', K, A( 1 ), N+1, INFO )
! 323: CALL ZHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE,
! 324: + A( 1 ), N+1 )
! 325: CALL ZTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1,
! 326: + A( K+1 ), N+1 )
! 327: CALL ZLAUUM( 'U', K, A( 0 ), N+1, INFO )
! 328: *
! 329: ELSE
! 330: *
! 331: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
! 332: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
! 333: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
! 334: *
! 335: CALL ZLAUUM( 'L', K, A( K+1 ), N+1, INFO )
! 336: CALL ZHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
! 337: + A( K+1 ), N+1 )
! 338: CALL ZTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1,
! 339: + A( 0 ), N+1 )
! 340: CALL ZLAUUM( 'U', K, A( K ), N+1, INFO )
! 341: *
! 342: END IF
! 343: *
! 344: ELSE
! 345: *
! 346: * N is even and TRANSR = 'C'
! 347: *
! 348: IF( LOWER ) THEN
! 349: *
! 350: * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
! 351: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
! 352: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
! 353: *
! 354: CALL ZLAUUM( 'U', K, A( K ), K, INFO )
! 355: CALL ZHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
! 356: + A( K ), K )
! 357: CALL ZTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K,
! 358: + A( K*( K+1 ) ), K )
! 359: CALL ZLAUUM( 'L', K, A( 0 ), K, INFO )
! 360: *
! 361: ELSE
! 362: *
! 363: * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
! 364: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
! 365: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
! 366: *
! 367: CALL ZLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
! 368: CALL ZHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE,
! 369: + A( K*( K+1 ) ), K )
! 370: CALL ZTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K,
! 371: + A( 0 ), K )
! 372: CALL ZLAUUM( 'L', K, A( K*K ), K, INFO )
! 373: *
! 374: END IF
! 375: *
! 376: END IF
! 377: *
! 378: END IF
! 379: *
! 380: RETURN
! 381: *
! 382: * End of ZPFTRI
! 383: *
! 384: END
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