Annotation of rpl/lapack/lapack/dsptrf.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: CHARACTER UPLO
! 10: INTEGER INFO, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: INTEGER IPIV( * )
! 14: DOUBLE PRECISION AP( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DSPTRF computes the factorization of a real symmetric matrix A stored
! 21: * in packed format using the Bunch-Kaufman diagonal pivoting method:
! 22: *
! 23: * A = U*D*U**T or A = L*D*L**T
! 24: *
! 25: * where U (or L) is a product of permutation and unit upper (lower)
! 26: * triangular matrices, and D is symmetric and block diagonal with
! 27: * 1-by-1 and 2-by-2 diagonal blocks.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * UPLO (input) CHARACTER*1
! 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: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 40: * On entry, the upper or lower triangle of the symmetric matrix
! 41: * A, packed columnwise in a linear array. The j-th column of A
! 42: * is stored in the array AP as follows:
! 43: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 44: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 45: *
! 46: * On exit, the block diagonal matrix D and the multipliers used
! 47: * to obtain the factor U or L, stored as a packed triangular
! 48: * matrix overwriting A (see below for further details).
! 49: *
! 50: * IPIV (output) INTEGER array, dimension (N)
! 51: * Details of the interchanges and the block structure of D.
! 52: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 53: * interchanged and D(k,k) is a 1-by-1 diagonal block.
! 54: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
! 55: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
! 56: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
! 57: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
! 58: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 59: *
! 60: * INFO (output) INTEGER
! 61: * = 0: successful exit
! 62: * < 0: if INFO = -i, the i-th argument had an illegal value
! 63: * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
! 64: * has been completed, but the block diagonal matrix D is
! 65: * exactly singular, and division by zero will occur if it
! 66: * is used to solve a system of equations.
! 67: *
! 68: * Further Details
! 69: * ===============
! 70: *
! 71: * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services
! 72: * Company
! 73: *
! 74: * If UPLO = 'U', then A = U*D*U', where
! 75: * U = P(n)*U(n)* ... *P(k)U(k)* ...,
! 76: * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
! 77: * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 78: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 79: * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
! 80: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 81: *
! 82: * ( I v 0 ) k-s
! 83: * U(k) = ( 0 I 0 ) s
! 84: * ( 0 0 I ) n-k
! 85: * k-s s n-k
! 86: *
! 87: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
! 88: * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
! 89: * and A(k,k), and v overwrites A(1:k-2,k-1:k).
! 90: *
! 91: * If UPLO = 'L', then A = L*D*L', where
! 92: * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
! 93: * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
! 94: * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 95: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 96: * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
! 97: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 98: *
! 99: * ( I 0 0 ) k-1
! 100: * L(k) = ( 0 I 0 ) s
! 101: * ( 0 v I ) n-k-s+1
! 102: * k-1 s n-k-s+1
! 103: *
! 104: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
! 105: * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
! 106: * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
! 107: *
! 108: * =====================================================================
! 109: *
! 110: * .. Parameters ..
! 111: DOUBLE PRECISION ZERO, ONE
! 112: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 113: DOUBLE PRECISION EIGHT, SEVTEN
! 114: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 115: * ..
! 116: * .. Local Scalars ..
! 117: LOGICAL UPPER
! 118: INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
! 119: $ KSTEP, KX, NPP
! 120: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
! 121: $ ROWMAX, T, WK, WKM1, WKP1
! 122: * ..
! 123: * .. External Functions ..
! 124: LOGICAL LSAME
! 125: INTEGER IDAMAX
! 126: EXTERNAL LSAME, IDAMAX
! 127: * ..
! 128: * .. External Subroutines ..
! 129: EXTERNAL DSCAL, DSPR, DSWAP, XERBLA
! 130: * ..
! 131: * .. Intrinsic Functions ..
! 132: INTRINSIC ABS, MAX, SQRT
! 133: * ..
! 134: * .. Executable Statements ..
! 135: *
! 136: * Test the input parameters.
! 137: *
! 138: INFO = 0
! 139: UPPER = LSAME( UPLO, 'U' )
! 140: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 141: INFO = -1
! 142: ELSE IF( N.LT.0 ) THEN
! 143: INFO = -2
! 144: END IF
! 145: IF( INFO.NE.0 ) THEN
! 146: CALL XERBLA( 'DSPTRF', -INFO )
! 147: RETURN
! 148: END IF
! 149: *
! 150: * Initialize ALPHA for use in choosing pivot block size.
! 151: *
! 152: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 153: *
! 154: IF( UPPER ) THEN
! 155: *
! 156: * Factorize A as U*D*U' using the upper triangle of A
! 157: *
! 158: * K is the main loop index, decreasing from N to 1 in steps of
! 159: * 1 or 2
! 160: *
! 161: K = N
! 162: KC = ( N-1 )*N / 2 + 1
! 163: 10 CONTINUE
! 164: KNC = KC
! 165: *
! 166: * If K < 1, exit from loop
! 167: *
! 168: IF( K.LT.1 )
! 169: $ GO TO 110
! 170: KSTEP = 1
! 171: *
! 172: * Determine rows and columns to be interchanged and whether
! 173: * a 1-by-1 or 2-by-2 pivot block will be used
! 174: *
! 175: ABSAKK = ABS( AP( KC+K-1 ) )
! 176: *
! 177: * IMAX is the row-index of the largest off-diagonal element in
! 178: * column K, and COLMAX is its absolute value
! 179: *
! 180: IF( K.GT.1 ) THEN
! 181: IMAX = IDAMAX( K-1, AP( KC ), 1 )
! 182: COLMAX = ABS( AP( KC+IMAX-1 ) )
! 183: ELSE
! 184: COLMAX = ZERO
! 185: END IF
! 186: *
! 187: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 188: *
! 189: * Column K is zero: set INFO and continue
! 190: *
! 191: IF( INFO.EQ.0 )
! 192: $ INFO = K
! 193: KP = K
! 194: ELSE
! 195: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 196: *
! 197: * no interchange, use 1-by-1 pivot block
! 198: *
! 199: KP = K
! 200: ELSE
! 201: *
! 202: * JMAX is the column-index of the largest off-diagonal
! 203: * element in row IMAX, and ROWMAX is its absolute value
! 204: *
! 205: ROWMAX = ZERO
! 206: JMAX = IMAX
! 207: KX = IMAX*( IMAX+1 ) / 2 + IMAX
! 208: DO 20 J = IMAX + 1, K
! 209: IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
! 210: ROWMAX = ABS( AP( KX ) )
! 211: JMAX = J
! 212: END IF
! 213: KX = KX + J
! 214: 20 CONTINUE
! 215: KPC = ( IMAX-1 )*IMAX / 2 + 1
! 216: IF( IMAX.GT.1 ) THEN
! 217: JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 )
! 218: ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) )
! 219: END IF
! 220: *
! 221: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 222: *
! 223: * no interchange, use 1-by-1 pivot block
! 224: *
! 225: KP = K
! 226: ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
! 227: *
! 228: * interchange rows and columns K and IMAX, use 1-by-1
! 229: * pivot block
! 230: *
! 231: KP = IMAX
! 232: ELSE
! 233: *
! 234: * interchange rows and columns K-1 and IMAX, use 2-by-2
! 235: * pivot block
! 236: *
! 237: KP = IMAX
! 238: KSTEP = 2
! 239: END IF
! 240: END IF
! 241: *
! 242: KK = K - KSTEP + 1
! 243: IF( KSTEP.EQ.2 )
! 244: $ KNC = KNC - K + 1
! 245: IF( KP.NE.KK ) THEN
! 246: *
! 247: * Interchange rows and columns KK and KP in the leading
! 248: * submatrix A(1:k,1:k)
! 249: *
! 250: CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
! 251: KX = KPC + KP - 1
! 252: DO 30 J = KP + 1, KK - 1
! 253: KX = KX + J - 1
! 254: T = AP( KNC+J-1 )
! 255: AP( KNC+J-1 ) = AP( KX )
! 256: AP( KX ) = T
! 257: 30 CONTINUE
! 258: T = AP( KNC+KK-1 )
! 259: AP( KNC+KK-1 ) = AP( KPC+KP-1 )
! 260: AP( KPC+KP-1 ) = T
! 261: IF( KSTEP.EQ.2 ) THEN
! 262: T = AP( KC+K-2 )
! 263: AP( KC+K-2 ) = AP( KC+KP-1 )
! 264: AP( KC+KP-1 ) = T
! 265: END IF
! 266: END IF
! 267: *
! 268: * Update the leading submatrix
! 269: *
! 270: IF( KSTEP.EQ.1 ) THEN
! 271: *
! 272: * 1-by-1 pivot block D(k): column k now holds
! 273: *
! 274: * W(k) = U(k)*D(k)
! 275: *
! 276: * where U(k) is the k-th column of U
! 277: *
! 278: * Perform a rank-1 update of A(1:k-1,1:k-1) as
! 279: *
! 280: * A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
! 281: *
! 282: R1 = ONE / AP( KC+K-1 )
! 283: CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
! 284: *
! 285: * Store U(k) in column k
! 286: *
! 287: CALL DSCAL( K-1, R1, AP( KC ), 1 )
! 288: ELSE
! 289: *
! 290: * 2-by-2 pivot block D(k): columns k and k-1 now hold
! 291: *
! 292: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 293: *
! 294: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 295: * of U
! 296: *
! 297: * Perform a rank-2 update of A(1:k-2,1:k-2) as
! 298: *
! 299: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
! 300: * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
! 301: *
! 302: IF( K.GT.2 ) THEN
! 303: *
! 304: D12 = AP( K-1+( K-1 )*K / 2 )
! 305: D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
! 306: D11 = AP( K+( K-1 )*K / 2 ) / D12
! 307: T = ONE / ( D11*D22-ONE )
! 308: D12 = T / D12
! 309: *
! 310: DO 50 J = K - 2, 1, -1
! 311: WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
! 312: $ AP( J+( K-1 )*K / 2 ) )
! 313: WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
! 314: $ AP( J+( K-2 )*( K-1 ) / 2 ) )
! 315: DO 40 I = J, 1, -1
! 316: AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
! 317: $ AP( I+( K-1 )*K / 2 )*WK -
! 318: $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
! 319: 40 CONTINUE
! 320: AP( J+( K-1 )*K / 2 ) = WK
! 321: AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
! 322: 50 CONTINUE
! 323: *
! 324: END IF
! 325: *
! 326: END IF
! 327: END IF
! 328: *
! 329: * Store details of the interchanges in IPIV
! 330: *
! 331: IF( KSTEP.EQ.1 ) THEN
! 332: IPIV( K ) = KP
! 333: ELSE
! 334: IPIV( K ) = -KP
! 335: IPIV( K-1 ) = -KP
! 336: END IF
! 337: *
! 338: * Decrease K and return to the start of the main loop
! 339: *
! 340: K = K - KSTEP
! 341: KC = KNC - K
! 342: GO TO 10
! 343: *
! 344: ELSE
! 345: *
! 346: * Factorize A as L*D*L' using the lower triangle of A
! 347: *
! 348: * K is the main loop index, increasing from 1 to N in steps of
! 349: * 1 or 2
! 350: *
! 351: K = 1
! 352: KC = 1
! 353: NPP = N*( N+1 ) / 2
! 354: 60 CONTINUE
! 355: KNC = KC
! 356: *
! 357: * If K > N, exit from loop
! 358: *
! 359: IF( K.GT.N )
! 360: $ GO TO 110
! 361: KSTEP = 1
! 362: *
! 363: * Determine rows and columns to be interchanged and whether
! 364: * a 1-by-1 or 2-by-2 pivot block will be used
! 365: *
! 366: ABSAKK = ABS( AP( KC ) )
! 367: *
! 368: * IMAX is the row-index of the largest off-diagonal element in
! 369: * column K, and COLMAX is its absolute value
! 370: *
! 371: IF( K.LT.N ) THEN
! 372: IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 )
! 373: COLMAX = ABS( AP( KC+IMAX-K ) )
! 374: ELSE
! 375: COLMAX = ZERO
! 376: END IF
! 377: *
! 378: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 379: *
! 380: * Column K is zero: set INFO and continue
! 381: *
! 382: IF( INFO.EQ.0 )
! 383: $ INFO = K
! 384: KP = K
! 385: ELSE
! 386: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 387: *
! 388: * no interchange, use 1-by-1 pivot block
! 389: *
! 390: KP = K
! 391: ELSE
! 392: *
! 393: * JMAX is the column-index of the largest off-diagonal
! 394: * element in row IMAX, and ROWMAX is its absolute value
! 395: *
! 396: ROWMAX = ZERO
! 397: KX = KC + IMAX - K
! 398: DO 70 J = K, IMAX - 1
! 399: IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN
! 400: ROWMAX = ABS( AP( KX ) )
! 401: JMAX = J
! 402: END IF
! 403: KX = KX + N - J
! 404: 70 CONTINUE
! 405: KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
! 406: IF( IMAX.LT.N ) THEN
! 407: JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 )
! 408: ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) )
! 409: END IF
! 410: *
! 411: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 412: *
! 413: * no interchange, use 1-by-1 pivot block
! 414: *
! 415: KP = K
! 416: ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
! 417: *
! 418: * interchange rows and columns K and IMAX, use 1-by-1
! 419: * pivot block
! 420: *
! 421: KP = IMAX
! 422: ELSE
! 423: *
! 424: * interchange rows and columns K+1 and IMAX, use 2-by-2
! 425: * pivot block
! 426: *
! 427: KP = IMAX
! 428: KSTEP = 2
! 429: END IF
! 430: END IF
! 431: *
! 432: KK = K + KSTEP - 1
! 433: IF( KSTEP.EQ.2 )
! 434: $ KNC = KNC + N - K + 1
! 435: IF( KP.NE.KK ) THEN
! 436: *
! 437: * Interchange rows and columns KK and KP in the trailing
! 438: * submatrix A(k:n,k:n)
! 439: *
! 440: IF( KP.LT.N )
! 441: $ CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
! 442: $ 1 )
! 443: KX = KNC + KP - KK
! 444: DO 80 J = KK + 1, KP - 1
! 445: KX = KX + N - J + 1
! 446: T = AP( KNC+J-KK )
! 447: AP( KNC+J-KK ) = AP( KX )
! 448: AP( KX ) = T
! 449: 80 CONTINUE
! 450: T = AP( KNC )
! 451: AP( KNC ) = AP( KPC )
! 452: AP( KPC ) = T
! 453: IF( KSTEP.EQ.2 ) THEN
! 454: T = AP( KC+1 )
! 455: AP( KC+1 ) = AP( KC+KP-K )
! 456: AP( KC+KP-K ) = T
! 457: END IF
! 458: END IF
! 459: *
! 460: * Update the trailing submatrix
! 461: *
! 462: IF( KSTEP.EQ.1 ) THEN
! 463: *
! 464: * 1-by-1 pivot block D(k): column k now holds
! 465: *
! 466: * W(k) = L(k)*D(k)
! 467: *
! 468: * where L(k) is the k-th column of L
! 469: *
! 470: IF( K.LT.N ) THEN
! 471: *
! 472: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 473: *
! 474: * A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
! 475: *
! 476: R1 = ONE / AP( KC )
! 477: CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
! 478: $ AP( KC+N-K+1 ) )
! 479: *
! 480: * Store L(k) in column K
! 481: *
! 482: CALL DSCAL( N-K, R1, AP( KC+1 ), 1 )
! 483: END IF
! 484: ELSE
! 485: *
! 486: * 2-by-2 pivot block D(k): columns K and K+1 now hold
! 487: *
! 488: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 489: *
! 490: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 491: * of L
! 492: *
! 493: IF( K.LT.N-1 ) THEN
! 494: *
! 495: * Perform a rank-2 update of A(k+2:n,k+2:n) as
! 496: *
! 497: * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
! 498: * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
! 499: *
! 500: D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
! 501: D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
! 502: D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
! 503: T = ONE / ( D11*D22-ONE )
! 504: D21 = T / D21
! 505: *
! 506: DO 100 J = K + 2, N
! 507: WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
! 508: $ AP( J+K*( 2*N-K-1 ) / 2 ) )
! 509: WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
! 510: $ AP( J+( K-1 )*( 2*N-K ) / 2 ) )
! 511: *
! 512: DO 90 I = J, N
! 513: AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
! 514: $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
! 515: $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
! 516: 90 CONTINUE
! 517: *
! 518: AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
! 519: AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
! 520: *
! 521: 100 CONTINUE
! 522: END IF
! 523: END IF
! 524: END IF
! 525: *
! 526: * Store details of the interchanges in IPIV
! 527: *
! 528: IF( KSTEP.EQ.1 ) THEN
! 529: IPIV( K ) = KP
! 530: ELSE
! 531: IPIV( K ) = -KP
! 532: IPIV( K+1 ) = -KP
! 533: END IF
! 534: *
! 535: * Increase K and return to the start of the main loop
! 536: *
! 537: K = K + KSTEP
! 538: KC = KNC + N - K + 2
! 539: GO TO 60
! 540: *
! 541: END IF
! 542: *
! 543: 110 CONTINUE
! 544: RETURN
! 545: *
! 546: * End of DSPTRF
! 547: *
! 548: END
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