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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SSPTRF( UPLO, N, AP, IPIV, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.3.1) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * -- April 2011 -- 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER INFO, N 00011 * .. 00012 * .. Array Arguments .. 00013 INTEGER IPIV( * ) 00014 REAL AP( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * SSPTRF computes the factorization of a real symmetric matrix A stored 00021 * in packed format using the Bunch-Kaufman diagonal pivoting method: 00022 * 00023 * A = U*D*U**T or A = L*D*L**T 00024 * 00025 * where U (or L) is a product of permutation and unit upper (lower) 00026 * triangular matrices, and D is symmetric and block diagonal with 00027 * 1-by-1 and 2-by-2 diagonal blocks. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * UPLO (input) CHARACTER*1 00033 * = 'U': Upper triangle of A is stored; 00034 * = 'L': Lower triangle of A is stored. 00035 * 00036 * N (input) INTEGER 00037 * The order of the matrix A. N >= 0. 00038 * 00039 * AP (input/output) REAL array, dimension (N*(N+1)/2) 00040 * On entry, the upper or lower triangle of the symmetric matrix 00041 * A, packed columnwise in a linear array. The j-th column of A 00042 * is stored in the array AP as follows: 00043 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00044 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00045 * 00046 * On exit, the block diagonal matrix D and the multipliers used 00047 * to obtain the factor U or L, stored as a packed triangular 00048 * matrix overwriting A (see below for further details). 00049 * 00050 * IPIV (output) INTEGER array, dimension (N) 00051 * Details of the interchanges and the block structure of D. 00052 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were 00053 * interchanged and D(k,k) is a 1-by-1 diagonal block. 00054 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 00055 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 00056 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 00057 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 00058 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00059 * 00060 * INFO (output) INTEGER 00061 * = 0: successful exit 00062 * < 0: if INFO = -i, the i-th argument had an illegal value 00063 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization 00064 * has been completed, but the block diagonal matrix D is 00065 * exactly singular, and division by zero will occur if it 00066 * is used to solve a system of equations. 00067 * 00068 * Further Details 00069 * =============== 00070 * 00071 * 5-96 - Based on modifications by J. Lewis, Boeing Computer Services 00072 * Company 00073 * 00074 * If UPLO = 'U', then A = U*D*U**T, where 00075 * U = P(n)*U(n)* ... *P(k)U(k)* ..., 00076 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to 00077 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00078 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00079 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such 00080 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00081 * 00082 * ( I v 0 ) k-s 00083 * U(k) = ( 0 I 0 ) s 00084 * ( 0 0 I ) n-k 00085 * k-s s n-k 00086 * 00087 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). 00088 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), 00089 * and A(k,k), and v overwrites A(1:k-2,k-1:k). 00090 * 00091 * If UPLO = 'L', then A = L*D*L**T, where 00092 * L = P(1)*L(1)* ... *P(k)*L(k)* ..., 00093 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to 00094 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00095 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00096 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such 00097 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00098 * 00099 * ( I 0 0 ) k-1 00100 * L(k) = ( 0 I 0 ) s 00101 * ( 0 v I ) n-k-s+1 00102 * k-1 s n-k-s+1 00103 * 00104 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). 00105 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), 00106 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). 00107 * 00108 * ===================================================================== 00109 * 00110 * .. Parameters .. 00111 REAL ZERO, ONE 00112 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00113 REAL EIGHT, SEVTEN 00114 PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) 00115 * .. 00116 * .. Local Scalars .. 00117 LOGICAL UPPER 00118 INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC, 00119 $ KSTEP, KX, NPP 00120 REAL ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1, 00121 $ ROWMAX, T, WK, WKM1, WKP1 00122 * .. 00123 * .. External Functions .. 00124 LOGICAL LSAME 00125 INTEGER ISAMAX 00126 EXTERNAL LSAME, ISAMAX 00127 * .. 00128 * .. External Subroutines .. 00129 EXTERNAL SSCAL, SSPR, SSWAP, XERBLA 00130 * .. 00131 * .. Intrinsic Functions .. 00132 INTRINSIC ABS, MAX, SQRT 00133 * .. 00134 * .. Executable Statements .. 00135 * 00136 * Test the input parameters. 00137 * 00138 INFO = 0 00139 UPPER = LSAME( UPLO, 'U' ) 00140 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00141 INFO = -1 00142 ELSE IF( N.LT.0 ) THEN 00143 INFO = -2 00144 END IF 00145 IF( INFO.NE.0 ) THEN 00146 CALL XERBLA( 'SSPTRF', -INFO ) 00147 RETURN 00148 END IF 00149 * 00150 * Initialize ALPHA for use in choosing pivot block size. 00151 * 00152 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 00153 * 00154 IF( UPPER ) THEN 00155 * 00156 * Factorize A as U*D*U**T using the upper triangle of A 00157 * 00158 * K is the main loop index, decreasing from N to 1 in steps of 00159 * 1 or 2 00160 * 00161 K = N 00162 KC = ( N-1 )*N / 2 + 1 00163 10 CONTINUE 00164 KNC = KC 00165 * 00166 * If K < 1, exit from loop 00167 * 00168 IF( K.LT.1 ) 00169 $ GO TO 110 00170 KSTEP = 1 00171 * 00172 * Determine rows and columns to be interchanged and whether 00173 * a 1-by-1 or 2-by-2 pivot block will be used 00174 * 00175 ABSAKK = ABS( AP( KC+K-1 ) ) 00176 * 00177 * IMAX is the row-index of the largest off-diagonal element in 00178 * column K, and COLMAX is its absolute value 00179 * 00180 IF( K.GT.1 ) THEN 00181 IMAX = ISAMAX( K-1, AP( KC ), 1 ) 00182 COLMAX = ABS( AP( KC+IMAX-1 ) ) 00183 ELSE 00184 COLMAX = ZERO 00185 END IF 00186 * 00187 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00188 * 00189 * Column K is zero: set INFO and continue 00190 * 00191 IF( INFO.EQ.0 ) 00192 $ INFO = K 00193 KP = K 00194 ELSE 00195 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00196 * 00197 * no interchange, use 1-by-1 pivot block 00198 * 00199 KP = K 00200 ELSE 00201 * 00202 ROWMAX = ZERO 00203 JMAX = IMAX 00204 KX = IMAX*( IMAX+1 ) / 2 + IMAX 00205 DO 20 J = IMAX + 1, K 00206 IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN 00207 ROWMAX = ABS( AP( KX ) ) 00208 JMAX = J 00209 END IF 00210 KX = KX + J 00211 20 CONTINUE 00212 KPC = ( IMAX-1 )*IMAX / 2 + 1 00213 IF( IMAX.GT.1 ) THEN 00214 JMAX = ISAMAX( IMAX-1, AP( KPC ), 1 ) 00215 ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) ) 00216 END IF 00217 * 00218 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00219 * 00220 * no interchange, use 1-by-1 pivot block 00221 * 00222 KP = K 00223 ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN 00224 * 00225 * interchange rows and columns K and IMAX, use 1-by-1 00226 * pivot block 00227 * 00228 KP = IMAX 00229 ELSE 00230 * 00231 * interchange rows and columns K-1 and IMAX, use 2-by-2 00232 * pivot block 00233 * 00234 KP = IMAX 00235 KSTEP = 2 00236 END IF 00237 END IF 00238 * 00239 KK = K - KSTEP + 1 00240 IF( KSTEP.EQ.2 ) 00241 $ KNC = KNC - K + 1 00242 IF( KP.NE.KK ) THEN 00243 * 00244 * Interchange rows and columns KK and KP in the leading 00245 * submatrix A(1:k,1:k) 00246 * 00247 CALL SSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 ) 00248 KX = KPC + KP - 1 00249 DO 30 J = KP + 1, KK - 1 00250 KX = KX + J - 1 00251 T = AP( KNC+J-1 ) 00252 AP( KNC+J-1 ) = AP( KX ) 00253 AP( KX ) = T 00254 30 CONTINUE 00255 T = AP( KNC+KK-1 ) 00256 AP( KNC+KK-1 ) = AP( KPC+KP-1 ) 00257 AP( KPC+KP-1 ) = T 00258 IF( KSTEP.EQ.2 ) THEN 00259 T = AP( KC+K-2 ) 00260 AP( KC+K-2 ) = AP( KC+KP-1 ) 00261 AP( KC+KP-1 ) = T 00262 END IF 00263 END IF 00264 * 00265 * Update the leading submatrix 00266 * 00267 IF( KSTEP.EQ.1 ) THEN 00268 * 00269 * 1-by-1 pivot block D(k): column k now holds 00270 * 00271 * W(k) = U(k)*D(k) 00272 * 00273 * where U(k) is the k-th column of U 00274 * 00275 * Perform a rank-1 update of A(1:k-1,1:k-1) as 00276 * 00277 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T 00278 * 00279 R1 = ONE / AP( KC+K-1 ) 00280 CALL SSPR( UPLO, K-1, -R1, AP( KC ), 1, AP ) 00281 * 00282 * Store U(k) in column k 00283 * 00284 CALL SSCAL( K-1, R1, AP( KC ), 1 ) 00285 ELSE 00286 * 00287 * 2-by-2 pivot block D(k): columns k and k-1 now hold 00288 * 00289 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 00290 * 00291 * where U(k) and U(k-1) are the k-th and (k-1)-th columns 00292 * of U 00293 * 00294 * Perform a rank-2 update of A(1:k-2,1:k-2) as 00295 * 00296 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T 00297 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T 00298 * 00299 IF( K.GT.2 ) THEN 00300 * 00301 D12 = AP( K-1+( K-1 )*K / 2 ) 00302 D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12 00303 D11 = AP( K+( K-1 )*K / 2 ) / D12 00304 T = ONE / ( D11*D22-ONE ) 00305 D12 = T / D12 00306 * 00307 DO 50 J = K - 2, 1, -1 00308 WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )- 00309 $ AP( J+( K-1 )*K / 2 ) ) 00310 WK = D12*( D22*AP( J+( K-1 )*K / 2 )- 00311 $ AP( J+( K-2 )*( K-1 ) / 2 ) ) 00312 DO 40 I = J, 1, -1 00313 AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) - 00314 $ AP( I+( K-1 )*K / 2 )*WK - 00315 $ AP( I+( K-2 )*( K-1 ) / 2 )*WKM1 00316 40 CONTINUE 00317 AP( J+( K-1 )*K / 2 ) = WK 00318 AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1 00319 50 CONTINUE 00320 * 00321 END IF 00322 * 00323 END IF 00324 END IF 00325 * 00326 * Store details of the interchanges in IPIV 00327 * 00328 IF( KSTEP.EQ.1 ) THEN 00329 IPIV( K ) = KP 00330 ELSE 00331 IPIV( K ) = -KP 00332 IPIV( K-1 ) = -KP 00333 END IF 00334 * 00335 * Decrease K and return to the start of the main loop 00336 * 00337 K = K - KSTEP 00338 KC = KNC - K 00339 GO TO 10 00340 * 00341 ELSE 00342 * 00343 * Factorize A as L*D*L**T using the lower triangle of A 00344 * 00345 * K is the main loop index, increasing from 1 to N in steps of 00346 * 1 or 2 00347 * 00348 K = 1 00349 KC = 1 00350 NPP = N*( N+1 ) / 2 00351 60 CONTINUE 00352 KNC = KC 00353 * 00354 * If K > N, exit from loop 00355 * 00356 IF( K.GT.N ) 00357 $ GO TO 110 00358 KSTEP = 1 00359 * 00360 * Determine rows and columns to be interchanged and whether 00361 * a 1-by-1 or 2-by-2 pivot block will be used 00362 * 00363 ABSAKK = ABS( AP( KC ) ) 00364 * 00365 * IMAX is the row-index of the largest off-diagonal element in 00366 * column K, and COLMAX is its absolute value 00367 * 00368 IF( K.LT.N ) THEN 00369 IMAX = K + ISAMAX( N-K, AP( KC+1 ), 1 ) 00370 COLMAX = ABS( AP( KC+IMAX-K ) ) 00371 ELSE 00372 COLMAX = ZERO 00373 END IF 00374 * 00375 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00376 * 00377 * Column K is zero: set INFO and continue 00378 * 00379 IF( INFO.EQ.0 ) 00380 $ INFO = K 00381 KP = K 00382 ELSE 00383 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00384 * 00385 * no interchange, use 1-by-1 pivot block 00386 * 00387 KP = K 00388 ELSE 00389 * 00390 * JMAX is the column-index of the largest off-diagonal 00391 * element in row IMAX, and ROWMAX is its absolute value 00392 * 00393 ROWMAX = ZERO 00394 KX = KC + IMAX - K 00395 DO 70 J = K, IMAX - 1 00396 IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN 00397 ROWMAX = ABS( AP( KX ) ) 00398 JMAX = J 00399 END IF 00400 KX = KX + N - J 00401 70 CONTINUE 00402 KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1 00403 IF( IMAX.LT.N ) THEN 00404 JMAX = IMAX + ISAMAX( N-IMAX, AP( KPC+1 ), 1 ) 00405 ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) ) 00406 END IF 00407 * 00408 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00409 * 00410 * no interchange, use 1-by-1 pivot block 00411 * 00412 KP = K 00413 ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN 00414 * 00415 * interchange rows and columns K and IMAX, use 1-by-1 00416 * pivot block 00417 * 00418 KP = IMAX 00419 ELSE 00420 * 00421 * interchange rows and columns K+1 and IMAX, use 2-by-2 00422 * pivot block 00423 * 00424 KP = IMAX 00425 KSTEP = 2 00426 END IF 00427 END IF 00428 * 00429 KK = K + KSTEP - 1 00430 IF( KSTEP.EQ.2 ) 00431 $ KNC = KNC + N - K + 1 00432 IF( KP.NE.KK ) THEN 00433 * 00434 * Interchange rows and columns KK and KP in the trailing 00435 * submatrix A(k:n,k:n) 00436 * 00437 IF( KP.LT.N ) 00438 $ CALL SSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ), 00439 $ 1 ) 00440 KX = KNC + KP - KK 00441 DO 80 J = KK + 1, KP - 1 00442 KX = KX + N - J + 1 00443 T = AP( KNC+J-KK ) 00444 AP( KNC+J-KK ) = AP( KX ) 00445 AP( KX ) = T 00446 80 CONTINUE 00447 T = AP( KNC ) 00448 AP( KNC ) = AP( KPC ) 00449 AP( KPC ) = T 00450 IF( KSTEP.EQ.2 ) THEN 00451 T = AP( KC+1 ) 00452 AP( KC+1 ) = AP( KC+KP-K ) 00453 AP( KC+KP-K ) = T 00454 END IF 00455 END IF 00456 * 00457 * Update the trailing submatrix 00458 * 00459 IF( KSTEP.EQ.1 ) THEN 00460 * 00461 * 1-by-1 pivot block D(k): column k now holds 00462 * 00463 * W(k) = L(k)*D(k) 00464 * 00465 * where L(k) is the k-th column of L 00466 * 00467 IF( K.LT.N ) THEN 00468 * 00469 * Perform a rank-1 update of A(k+1:n,k+1:n) as 00470 * 00471 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T 00472 * 00473 R1 = ONE / AP( KC ) 00474 CALL SSPR( UPLO, N-K, -R1, AP( KC+1 ), 1, 00475 $ AP( KC+N-K+1 ) ) 00476 * 00477 * Store L(k) in column K 00478 * 00479 CALL SSCAL( N-K, R1, AP( KC+1 ), 1 ) 00480 END IF 00481 ELSE 00482 * 00483 * 2-by-2 pivot block D(k): columns K and K+1 now hold 00484 * 00485 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 00486 * 00487 * where L(k) and L(k+1) are the k-th and (k+1)-th columns 00488 * of L 00489 * 00490 IF( K.LT.N-1 ) THEN 00491 * 00492 * Perform a rank-2 update of A(k+2:n,k+2:n) as 00493 * 00494 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T 00495 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T 00496 * 00497 * where L(k) and L(k+1) are the k-th and (k+1)-th 00498 * columns of L 00499 * 00500 D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 ) 00501 D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21 00502 D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21 00503 T = ONE / ( D11*D22-ONE ) 00504 D21 = T / D21 00505 * 00506 DO 100 J = K + 2, N 00507 WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )- 00508 $ AP( J+K*( 2*N-K-1 ) / 2 ) ) 00509 WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )- 00510 $ AP( J+( K-1 )*( 2*N-K ) / 2 ) ) 00511 * 00512 DO 90 I = J, N 00513 AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )* 00514 $ ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) / 00515 $ 2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1 00516 90 CONTINUE 00517 * 00518 AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK 00519 AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1 00520 * 00521 100 CONTINUE 00522 END IF 00523 END IF 00524 END IF 00525 * 00526 * Store details of the interchanges in IPIV 00527 * 00528 IF( KSTEP.EQ.1 ) THEN 00529 IPIV( K ) = KP 00530 ELSE 00531 IPIV( K ) = -KP 00532 IPIV( K+1 ) = -KP 00533 END IF 00534 * 00535 * Increase K and return to the start of the main loop 00536 * 00537 K = K + KSTEP 00538 KC = KNC + N - K + 2 00539 GO TO 60 00540 * 00541 END IF 00542 * 00543 110 CONTINUE 00544 RETURN 00545 * 00546 * End of SSPTRF 00547 * 00548 END
1.7.3 
