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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE ZSPTRF( UPLO, N, AP, IPIV, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER INFO, N 00011 * .. 00012 * .. Array Arguments .. 00013 INTEGER IPIV( * ) 00014 COMPLEX*16 AP( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * ZSPTRF computes the factorization of a complex symmetric matrix A 00021 * stored in packed format using the Bunch-Kaufman diagonal pivoting 00022 * method: 00023 * 00024 * A = U*D*U**T or A = L*D*L**T 00025 * 00026 * where U (or L) is a product of permutation and unit upper (lower) 00027 * triangular matrices, and D is symmetric and block diagonal with 00028 * 1-by-1 and 2-by-2 diagonal blocks. 00029 * 00030 * Arguments 00031 * ========= 00032 * 00033 * UPLO (input) CHARACTER*1 00034 * = 'U': Upper triangle of A is stored; 00035 * = 'L': Lower triangle of A is stored. 00036 * 00037 * N (input) INTEGER 00038 * The order of the matrix A. N >= 0. 00039 * 00040 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) 00041 * On entry, the upper or lower triangle of the symmetric matrix 00042 * A, packed columnwise in a linear array. The j-th column of A 00043 * is stored in the array AP as follows: 00044 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00045 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00046 * 00047 * On exit, the block diagonal matrix D and the multipliers used 00048 * to obtain the factor U or L, stored as a packed triangular 00049 * matrix overwriting A (see below for further details). 00050 * 00051 * IPIV (output) INTEGER array, dimension (N) 00052 * Details of the interchanges and the block structure of D. 00053 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were 00054 * interchanged and D(k,k) is a 1-by-1 diagonal block. 00055 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 00056 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 00057 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 00058 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 00059 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00060 * 00061 * INFO (output) INTEGER 00062 * = 0: successful exit 00063 * < 0: if INFO = -i, the i-th argument had an illegal value 00064 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization 00065 * has been completed, but the block diagonal matrix D is 00066 * exactly singular, and division by zero will occur if it 00067 * is used to solve a system of equations. 00068 * 00069 * Further Details 00070 * =============== 00071 * 00072 * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services 00073 * Company 00074 * 00075 * If UPLO = 'U', then A = U*D*U', where 00076 * U = P(n)*U(n)* ... *P(k)U(k)* ..., 00077 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to 00078 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00079 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00080 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such 00081 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00082 * 00083 * ( I v 0 ) k-s 00084 * U(k) = ( 0 I 0 ) s 00085 * ( 0 0 I ) n-k 00086 * k-s s n-k 00087 * 00088 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). 00089 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), 00090 * and A(k,k), and v overwrites A(1:k-2,k-1:k). 00091 * 00092 * If UPLO = 'L', then A = L*D*L', where 00093 * L = P(1)*L(1)* ... *P(k)*L(k)* ..., 00094 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to 00095 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00096 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00097 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such 00098 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00099 * 00100 * ( I 0 0 ) k-1 00101 * L(k) = ( 0 I 0 ) s 00102 * ( 0 v I ) n-k-s+1 00103 * k-1 s n-k-s+1 00104 * 00105 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). 00106 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), 00107 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). 00108 * 00109 * ===================================================================== 00110 * 00111 * .. Parameters .. 00112 DOUBLE PRECISION ZERO, ONE 00113 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00114 DOUBLE PRECISION EIGHT, SEVTEN 00115 PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) 00116 COMPLEX*16 CONE 00117 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 00118 * .. 00119 * .. Local Scalars .. 00120 LOGICAL UPPER 00121 INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC, 00122 $ KSTEP, KX, NPP 00123 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX 00124 COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, ZDUM 00125 * .. 00126 * .. External Functions .. 00127 LOGICAL LSAME 00128 INTEGER IZAMAX 00129 EXTERNAL LSAME, IZAMAX 00130 * .. 00131 * .. External Subroutines .. 00132 EXTERNAL XERBLA, ZSCAL, ZSPR, ZSWAP 00133 * .. 00134 * .. Intrinsic Functions .. 00135 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT 00136 * .. 00137 * .. Statement Functions .. 00138 DOUBLE PRECISION CABS1 00139 * .. 00140 * .. Statement Function definitions .. 00141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00142 * .. 00143 * .. Executable Statements .. 00144 * 00145 * Test the input parameters. 00146 * 00147 INFO = 0 00148 UPPER = LSAME( UPLO, 'U' ) 00149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00150 INFO = -1 00151 ELSE IF( N.LT.0 ) THEN 00152 INFO = -2 00153 END IF 00154 IF( INFO.NE.0 ) THEN 00155 CALL XERBLA( 'ZSPTRF', -INFO ) 00156 RETURN 00157 END IF 00158 * 00159 * Initialize ALPHA for use in choosing pivot block size. 00160 * 00161 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 00162 * 00163 IF( UPPER ) THEN 00164 * 00165 * Factorize A as U*D*U' using the upper triangle of A 00166 * 00167 * K is the main loop index, decreasing from N to 1 in steps of 00168 * 1 or 2 00169 * 00170 K = N 00171 KC = ( N-1 )*N / 2 + 1 00172 10 CONTINUE 00173 KNC = KC 00174 * 00175 * If K < 1, exit from loop 00176 * 00177 IF( K.LT.1 ) 00178 $ GO TO 110 00179 KSTEP = 1 00180 * 00181 * Determine rows and columns to be interchanged and whether 00182 * a 1-by-1 or 2-by-2 pivot block will be used 00183 * 00184 ABSAKK = CABS1( AP( KC+K-1 ) ) 00185 * 00186 * IMAX is the row-index of the largest off-diagonal element in 00187 * column K, and COLMAX is its absolute value 00188 * 00189 IF( K.GT.1 ) THEN 00190 IMAX = IZAMAX( K-1, AP( KC ), 1 ) 00191 COLMAX = CABS1( AP( KC+IMAX-1 ) ) 00192 ELSE 00193 COLMAX = ZERO 00194 END IF 00195 * 00196 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00197 * 00198 * Column K is zero: set INFO and continue 00199 * 00200 IF( INFO.EQ.0 ) 00201 $ INFO = K 00202 KP = K 00203 ELSE 00204 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00205 * 00206 * no interchange, use 1-by-1 pivot block 00207 * 00208 KP = K 00209 ELSE 00210 * 00211 * JMAX is the column-index of the largest off-diagonal 00212 * element in row IMAX, and ROWMAX is its absolute value 00213 * 00214 ROWMAX = ZERO 00215 JMAX = IMAX 00216 KX = IMAX*( IMAX+1 ) / 2 + IMAX 00217 DO 20 J = IMAX + 1, K 00218 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN 00219 ROWMAX = CABS1( AP( KX ) ) 00220 JMAX = J 00221 END IF 00222 KX = KX + J 00223 20 CONTINUE 00224 KPC = ( IMAX-1 )*IMAX / 2 + 1 00225 IF( IMAX.GT.1 ) THEN 00226 JMAX = IZAMAX( IMAX-1, AP( KPC ), 1 ) 00227 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-1 ) ) ) 00228 END IF 00229 * 00230 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00231 * 00232 * no interchange, use 1-by-1 pivot block 00233 * 00234 KP = K 00235 ELSE IF( CABS1( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN 00236 * 00237 * interchange rows and columns K and IMAX, use 1-by-1 00238 * pivot block 00239 * 00240 KP = IMAX 00241 ELSE 00242 * 00243 * interchange rows and columns K-1 and IMAX, use 2-by-2 00244 * pivot block 00245 * 00246 KP = IMAX 00247 KSTEP = 2 00248 END IF 00249 END IF 00250 * 00251 KK = K - KSTEP + 1 00252 IF( KSTEP.EQ.2 ) 00253 $ KNC = KNC - K + 1 00254 IF( KP.NE.KK ) THEN 00255 * 00256 * Interchange rows and columns KK and KP in the leading 00257 * submatrix A(1:k,1:k) 00258 * 00259 CALL ZSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 ) 00260 KX = KPC + KP - 1 00261 DO 30 J = KP + 1, KK - 1 00262 KX = KX + J - 1 00263 T = AP( KNC+J-1 ) 00264 AP( KNC+J-1 ) = AP( KX ) 00265 AP( KX ) = T 00266 30 CONTINUE 00267 T = AP( KNC+KK-1 ) 00268 AP( KNC+KK-1 ) = AP( KPC+KP-1 ) 00269 AP( KPC+KP-1 ) = T 00270 IF( KSTEP.EQ.2 ) THEN 00271 T = AP( KC+K-2 ) 00272 AP( KC+K-2 ) = AP( KC+KP-1 ) 00273 AP( KC+KP-1 ) = T 00274 END IF 00275 END IF 00276 * 00277 * Update the leading submatrix 00278 * 00279 IF( KSTEP.EQ.1 ) THEN 00280 * 00281 * 1-by-1 pivot block D(k): column k now holds 00282 * 00283 * W(k) = U(k)*D(k) 00284 * 00285 * where U(k) is the k-th column of U 00286 * 00287 * Perform a rank-1 update of A(1:k-1,1:k-1) as 00288 * 00289 * A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' 00290 * 00291 R1 = CONE / AP( KC+K-1 ) 00292 CALL ZSPR( UPLO, K-1, -R1, AP( KC ), 1, AP ) 00293 * 00294 * Store U(k) in column k 00295 * 00296 CALL ZSCAL( K-1, R1, AP( KC ), 1 ) 00297 ELSE 00298 * 00299 * 2-by-2 pivot block D(k): columns k and k-1 now hold 00300 * 00301 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 00302 * 00303 * where U(k) and U(k-1) are the k-th and (k-1)-th columns 00304 * of U 00305 * 00306 * Perform a rank-2 update of A(1:k-2,1:k-2) as 00307 * 00308 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' 00309 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' 00310 * 00311 IF( K.GT.2 ) THEN 00312 * 00313 D12 = AP( K-1+( K-1 )*K / 2 ) 00314 D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12 00315 D11 = AP( K+( K-1 )*K / 2 ) / D12 00316 T = CONE / ( D11*D22-CONE ) 00317 D12 = T / D12 00318 * 00319 DO 50 J = K - 2, 1, -1 00320 WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )- 00321 $ AP( J+( K-1 )*K / 2 ) ) 00322 WK = D12*( D22*AP( J+( K-1 )*K / 2 )- 00323 $ AP( J+( K-2 )*( K-1 ) / 2 ) ) 00324 DO 40 I = J, 1, -1 00325 AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) - 00326 $ AP( I+( K-1 )*K / 2 )*WK - 00327 $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1 00328 40 CONTINUE 00329 AP( J+( K-1 )*K / 2 ) = WK 00330 AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1 00331 50 CONTINUE 00332 * 00333 END IF 00334 END IF 00335 END IF 00336 * 00337 * Store details of the interchanges in IPIV 00338 * 00339 IF( KSTEP.EQ.1 ) THEN 00340 IPIV( K ) = KP 00341 ELSE 00342 IPIV( K ) = -KP 00343 IPIV( K-1 ) = -KP 00344 END IF 00345 * 00346 * Decrease K and return to the start of the main loop 00347 * 00348 K = K - KSTEP 00349 KC = KNC - K 00350 GO TO 10 00351 * 00352 ELSE 00353 * 00354 * Factorize A as L*D*L' using the lower triangle of A 00355 * 00356 * K is the main loop index, increasing from 1 to N in steps of 00357 * 1 or 2 00358 * 00359 K = 1 00360 KC = 1 00361 NPP = N*( N+1 ) / 2 00362 60 CONTINUE 00363 KNC = KC 00364 * 00365 * If K > N, exit from loop 00366 * 00367 IF( K.GT.N ) 00368 $ GO TO 110 00369 KSTEP = 1 00370 * 00371 * Determine rows and columns to be interchanged and whether 00372 * a 1-by-1 or 2-by-2 pivot block will be used 00373 * 00374 ABSAKK = CABS1( AP( KC ) ) 00375 * 00376 * IMAX is the row-index of the largest off-diagonal element in 00377 * column K, and COLMAX is its absolute value 00378 * 00379 IF( K.LT.N ) THEN 00380 IMAX = K + IZAMAX( N-K, AP( KC+1 ), 1 ) 00381 COLMAX = CABS1( AP( KC+IMAX-K ) ) 00382 ELSE 00383 COLMAX = ZERO 00384 END IF 00385 * 00386 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00387 * 00388 * Column K is zero: set INFO and continue 00389 * 00390 IF( INFO.EQ.0 ) 00391 $ INFO = K 00392 KP = K 00393 ELSE 00394 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00395 * 00396 * no interchange, use 1-by-1 pivot block 00397 * 00398 KP = K 00399 ELSE 00400 * 00401 * JMAX is the column-index of the largest off-diagonal 00402 * element in row IMAX, and ROWMAX is its absolute value 00403 * 00404 ROWMAX = ZERO 00405 KX = KC + IMAX - K 00406 DO 70 J = K, IMAX - 1 00407 IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN 00408 ROWMAX = CABS1( AP( KX ) ) 00409 JMAX = J 00410 END IF 00411 KX = KX + N - J 00412 70 CONTINUE 00413 KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1 00414 IF( IMAX.LT.N ) THEN 00415 JMAX = IMAX + IZAMAX( N-IMAX, AP( KPC+1 ), 1 ) 00416 ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-IMAX ) ) ) 00417 END IF 00418 * 00419 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00420 * 00421 * no interchange, use 1-by-1 pivot block 00422 * 00423 KP = K 00424 ELSE IF( CABS1( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN 00425 * 00426 * interchange rows and columns K and IMAX, use 1-by-1 00427 * pivot block 00428 * 00429 KP = IMAX 00430 ELSE 00431 * 00432 * interchange rows and columns K+1 and IMAX, use 2-by-2 00433 * pivot block 00434 * 00435 KP = IMAX 00436 KSTEP = 2 00437 END IF 00438 END IF 00439 * 00440 KK = K + KSTEP - 1 00441 IF( KSTEP.EQ.2 ) 00442 $ KNC = KNC + N - K + 1 00443 IF( KP.NE.KK ) THEN 00444 * 00445 * Interchange rows and columns KK and KP in the trailing 00446 * submatrix A(k:n,k:n) 00447 * 00448 IF( KP.LT.N ) 00449 $ CALL ZSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ), 00450 $ 1 ) 00451 KX = KNC + KP - KK 00452 DO 80 J = KK + 1, KP - 1 00453 KX = KX + N - J + 1 00454 T = AP( KNC+J-KK ) 00455 AP( KNC+J-KK ) = AP( KX ) 00456 AP( KX ) = T 00457 80 CONTINUE 00458 T = AP( KNC ) 00459 AP( KNC ) = AP( KPC ) 00460 AP( KPC ) = T 00461 IF( KSTEP.EQ.2 ) THEN 00462 T = AP( KC+1 ) 00463 AP( KC+1 ) = AP( KC+KP-K ) 00464 AP( KC+KP-K ) = T 00465 END IF 00466 END IF 00467 * 00468 * Update the trailing submatrix 00469 * 00470 IF( KSTEP.EQ.1 ) THEN 00471 * 00472 * 1-by-1 pivot block D(k): column k now holds 00473 * 00474 * W(k) = L(k)*D(k) 00475 * 00476 * where L(k) is the k-th column of L 00477 * 00478 IF( K.LT.N ) THEN 00479 * 00480 * Perform a rank-1 update of A(k+1:n,k+1:n) as 00481 * 00482 * A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' 00483 * 00484 R1 = CONE / AP( KC ) 00485 CALL ZSPR( UPLO, N-K, -R1, AP( KC+1 ), 1, 00486 $ AP( KC+N-K+1 ) ) 00487 * 00488 * Store L(k) in column K 00489 * 00490 CALL ZSCAL( N-K, R1, AP( KC+1 ), 1 ) 00491 END IF 00492 ELSE 00493 * 00494 * 2-by-2 pivot block D(k): columns K and K+1 now hold 00495 * 00496 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 00497 * 00498 * where L(k) and L(k+1) are the k-th and (k+1)-th columns 00499 * of L 00500 * 00501 IF( K.LT.N-1 ) THEN 00502 * 00503 * Perform a rank-2 update of A(k+2:n,k+2:n) as 00504 * 00505 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' 00506 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' 00507 * 00508 * where L(k) and L(k+1) are the k-th and (k+1)-th 00509 * columns of L 00510 * 00511 D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 ) 00512 D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21 00513 D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21 00514 T = CONE / ( D11*D22-CONE ) 00515 D21 = T / D21 00516 * 00517 DO 100 J = K + 2, N 00518 WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )- 00519 $ AP( J+K*( 2*N-K-1 ) / 2 ) ) 00520 WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )- 00521 $ AP( J+( K-1 )*( 2*N-K ) / 2 ) ) 00522 DO 90 I = J, N 00523 AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )* 00524 $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) / 00525 $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1 00526 90 CONTINUE 00527 AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK 00528 AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1 00529 100 CONTINUE 00530 END IF 00531 END IF 00532 END IF 00533 * 00534 * Store details of the interchanges in IPIV 00535 * 00536 IF( KSTEP.EQ.1 ) THEN 00537 IPIV( K ) = KP 00538 ELSE 00539 IPIV( K ) = -KP 00540 IPIV( K+1 ) = -KP 00541 END IF 00542 * 00543 * Increase K and return to the start of the main loop 00544 * 00545 K = K + KSTEP 00546 KC = KNC + N - K + 2 00547 GO TO 60 00548 * 00549 END IF 00550 * 00551 110 CONTINUE 00552 RETURN 00553 * 00554 * End of ZSPTRF 00555 * 00556 END
1.7.3 
