#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int clags2_(logical *upper, real *a1, complex *a2, real *a3, real *b1, complex *b2, real *b3, real *csu, complex *snu, real *csv, complex *snv, real *csq, complex *snq) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such that if ( UPPER ) then U'*A*Q = U'*( A1 A2 )*Q = ( x 0 ) ( 0 A3 ) ( x x ) and V'*B*Q = V'*( B1 B2 )*Q = ( x 0 ) ( 0 B3 ) ( x x ) or if ( .NOT.UPPER ) then U'*A*Q = U'*( A1 0 )*Q = ( x x ) ( A2 A3 ) ( 0 x ) and V'*B*Q = V'*( B1 0 )*Q = ( x x ) ( B2 B3 ) ( 0 x ) where U = ( CSU SNU ), V = ( CSV SNV ), ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV ) Q = ( CSQ SNQ ) ( -CONJG(SNQ) CSQ ) Z' denotes the conjugate transpose of Z. The rows of the transformed A and B are parallel. Moreover, if the input 2-by-2 matrix A is not zero, then the transformed (1,1) entry of A is not zero. If the input matrices A and B are both not zero, then the transformed (2,2) element of B is not zero, except when the first rows of input A and B are parallel and the second rows are zero. Arguments ========= UPPER (input) LOGICAL = .TRUE.: the input matrices A and B are upper triangular. = .FALSE.: the input matrices A and B are lower triangular. A1 (input) REAL A2 (input) COMPLEX A3 (input) REAL On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A. B1 (input) REAL B2 (input) COMPLEX B3 (input) REAL On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B. CSU (output) REAL SNU (output) COMPLEX The desired unitary matrix U. CSV (output) REAL SNV (output) COMPLEX The desired unitary matrix V. CSQ (output) REAL SNQ (output) COMPLEX The desired unitary matrix Q. ===================================================================== */ /* System generated locals */ real r__1, r__2, r__3, r__4, r__5, r__6, r__7, r__8; complex q__1, q__2, q__3, q__4, q__5; /* Builtin functions */ double c_abs(complex *), r_imag(complex *); void r_cnjg(complex *, complex *); /* Local variables */ static real a; static complex b, c__; static real d__; static complex r__, d1; static real s1, s2, fb, fc; static complex ua11, ua12, ua21, ua22, vb11, vb12, vb21, vb22; static real csl, csr, snl, snr, aua11, aua12, aua21, aua22, avb11, avb12, avb21, avb22, ua11r, ua22r, vb11r, vb22r; extern /* Subroutine */ int slasv2_(real *, real *, real *, real *, real * , real *, real *, real *, real *), clartg_(complex *, complex *, real *, complex *, complex *); if (*upper) { /* Input matrices A and B are upper triangular matrices Form matrix C = A*adj(B) = ( a b ) ( 0 d ) */ a = *a1 * *b3; d__ = *a3 * *b1; q__2.r = *b1 * a2->r, q__2.i = *b1 * a2->i; q__3.r = *a1 * b2->r, q__3.i = *a1 * b2->i; q__1.r = q__2.r - q__3.r, q__1.i = q__2.i - q__3.i; b.r = q__1.r, b.i = q__1.i; fb = c_abs(&b); /* Transform complex 2-by-2 matrix C to real matrix by unitary diagonal matrix diag(1,D1). */ d1.r = 1.f, d1.i = 0.f; if (fb != 0.f) { q__1.r = b.r / fb, q__1.i = b.i / fb; d1.r = q__1.r, d1.i = q__1.i; } /* The SVD of real 2 by 2 triangular C ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 ) ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T ) */ slasv2_(&a, &fb, &d__, &s1, &s2, &snr, &csr, &snl, &csl); if (dabs(csl) >= dabs(snl) || dabs(csr) >= dabs(snr)) { /* Compute the (1,1) and (1,2) elements of U'*A and V'*B, and (1,2) element of |U|'*|A| and |V|'*|B|. */ ua11r = csl * *a1; q__2.r = csl * a2->r, q__2.i = csl * a2->i; q__4.r = snl * d1.r, q__4.i = snl * d1.i; q__3.r = *a3 * q__4.r, q__3.i = *a3 * q__4.i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; ua12.r = q__1.r, ua12.i = q__1.i; vb11r = csr * *b1; q__2.r = csr * b2->r, q__2.i = csr * b2->i; q__4.r = snr * d1.r, q__4.i = snr * d1.i; q__3.r = *b3 * q__4.r, q__3.i = *b3 * q__4.i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; vb12.r = q__1.r, vb12.i = q__1.i; aua12 = dabs(csl) * ((r__1 = a2->r, dabs(r__1)) + (r__2 = r_imag( a2), dabs(r__2))) + dabs(snl) * dabs(*a3); avb12 = dabs(csr) * ((r__1 = b2->r, dabs(r__1)) + (r__2 = r_imag( b2), dabs(r__2))) + dabs(snr) * dabs(*b3); /* zero (1,2) elements of U'*A and V'*B */ if (dabs(ua11r) + ((r__1 = ua12.r, dabs(r__1)) + (r__2 = r_imag(& ua12), dabs(r__2))) == 0.f) { q__2.r = vb11r, q__2.i = 0.f; q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &vb12); clartg_(&q__1, &q__3, csq, snq, &r__); } else if (dabs(vb11r) + ((r__1 = vb12.r, dabs(r__1)) + (r__2 = r_imag(&vb12), dabs(r__2))) == 0.f) { q__2.r = ua11r, q__2.i = 0.f; q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &ua12); clartg_(&q__1, &q__3, csq, snq, &r__); } else if (aua12 / (dabs(ua11r) + ((r__1 = ua12.r, dabs(r__1)) + ( r__2 = r_imag(&ua12), dabs(r__2)))) <= avb12 / (dabs( vb11r) + ((r__3 = vb12.r, dabs(r__3)) + (r__4 = r_imag(& vb12), dabs(r__4))))) { q__2.r = ua11r, q__2.i = 0.f; q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &ua12); clartg_(&q__1, &q__3, csq, snq, &r__); } else { q__2.r = vb11r, q__2.i = 0.f; q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &vb12); clartg_(&q__1, &q__3, csq, snq, &r__); } *csu = csl; q__2.r = -d1.r, q__2.i = -d1.i; q__1.r = snl * q__2.r, q__1.i = snl * q__2.i; snu->r = q__1.r, snu->i = q__1.i; *csv = csr; q__2.r = -d1.r, q__2.i = -d1.i; q__1.r = snr * q__2.r, q__1.i = snr * q__2.i; snv->r = q__1.r, snv->i = q__1.i; } else { /* Compute the (2,1) and (2,2) elements of U'*A and V'*B, and (2,2) element of |U|'*|A| and |V|'*|B|. */ r_cnjg(&q__4, &d1); q__3.r = -q__4.r, q__3.i = -q__4.i; q__2.r = snl * q__3.r, q__2.i = snl * q__3.i; q__1.r = *a1 * q__2.r, q__1.i = *a1 * q__2.i; ua21.r = q__1.r, ua21.i = q__1.i; r_cnjg(&q__5, &d1); q__4.r = -q__5.r, q__4.i = -q__5.i; q__3.r = snl * q__4.r, q__3.i = snl * q__4.i; q__2.r = q__3.r * a2->r - q__3.i * a2->i, q__2.i = q__3.r * a2->i + q__3.i * a2->r; r__1 = csl * *a3; q__1.r = q__2.r + r__1, q__1.i = q__2.i; ua22.r = q__1.r, ua22.i = q__1.i; r_cnjg(&q__4, &d1); q__3.r = -q__4.r, q__3.i = -q__4.i; q__2.r = snr * q__3.r, q__2.i = snr * q__3.i; q__1.r = *b1 * q__2.r, q__1.i = *b1 * q__2.i; vb21.r = q__1.r, vb21.i = q__1.i; r_cnjg(&q__5, &d1); q__4.r = -q__5.r, q__4.i = -q__5.i; q__3.r = snr * q__4.r, q__3.i = snr * q__4.i; q__2.r = q__3.r * b2->r - q__3.i * b2->i, q__2.i = q__3.r * b2->i + q__3.i * b2->r; r__1 = csr * *b3; q__1.r = q__2.r + r__1, q__1.i = q__2.i; vb22.r = q__1.r, vb22.i = q__1.i; aua22 = dabs(snl) * ((r__1 = a2->r, dabs(r__1)) + (r__2 = r_imag( a2), dabs(r__2))) + dabs(csl) * dabs(*a3); avb22 = dabs(snr) * ((r__1 = b2->r, dabs(r__1)) + (r__2 = r_imag( b2), dabs(r__2))) + dabs(csr) * dabs(*b3); /* zero (2,2) elements of U'*A and V'*B, and then swap. */ if ((r__1 = ua21.r, dabs(r__1)) + (r__2 = r_imag(&ua21), dabs( r__2)) + ((r__3 = ua22.r, dabs(r__3)) + (r__4 = r_imag(& ua22), dabs(r__4))) == 0.f) { r_cnjg(&q__2, &vb21); q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &vb22); clartg_(&q__1, &q__3, csq, snq, &r__); } else if ((r__1 = vb21.r, dabs(r__1)) + (r__2 = r_imag(&vb21), dabs(r__2)) + c_abs(&vb22) == 0.f) { r_cnjg(&q__2, &ua21); q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &ua22); clartg_(&q__1, &q__3, csq, snq, &r__); } else if (aua22 / ((r__1 = ua21.r, dabs(r__1)) + (r__2 = r_imag(& ua21), dabs(r__2)) + ((r__3 = ua22.r, dabs(r__3)) + (r__4 = r_imag(&ua22), dabs(r__4)))) <= avb22 / ((r__5 = vb21.r, dabs(r__5)) + (r__6 = r_imag(&vb21), dabs(r__6)) + (( r__7 = vb22.r, dabs(r__7)) + (r__8 = r_imag(&vb22), dabs( r__8))))) { r_cnjg(&q__2, &ua21); q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &ua22); clartg_(&q__1, &q__3, csq, snq, &r__); } else { r_cnjg(&q__2, &vb21); q__1.r = -q__2.r, q__1.i = -q__2.i; r_cnjg(&q__3, &vb22); clartg_(&q__1, &q__3, csq, snq, &r__); } *csu = snl; q__1.r = csl * d1.r, q__1.i = csl * d1.i; snu->r = q__1.r, snu->i = q__1.i; *csv = snr; q__1.r = csr * d1.r, q__1.i = csr * d1.i; snv->r = q__1.r, snv->i = q__1.i; } } else { /* Input matrices A and B are lower triangular matrices Form matrix C = A*adj(B) = ( a 0 ) ( c d ) */ a = *a1 * *b3; d__ = *a3 * *b1; q__2.r = *b3 * a2->r, q__2.i = *b3 * a2->i; q__3.r = *a3 * b2->r, q__3.i = *a3 * b2->i; q__1.r = q__2.r - q__3.r, q__1.i = q__2.i - q__3.i; c__.r = q__1.r, c__.i = q__1.i; fc = c_abs(&c__); /* Transform complex 2-by-2 matrix C to real matrix by unitary diagonal matrix diag(d1,1). */ d1.r = 1.f, d1.i = 0.f; if (fc != 0.f) { q__1.r = c__.r / fc, q__1.i = c__.i / fc; d1.r = q__1.r, d1.i = q__1.i; } /* The SVD of real 2 by 2 triangular C ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 ) ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T ) */ slasv2_(&a, &fc, &d__, &s1, &s2, &snr, &csr, &snl, &csl); if (dabs(csr) >= dabs(snr) || dabs(csl) >= dabs(snl)) { /* Compute the (2,1) and (2,2) elements of U'*A and V'*B, and (2,1) element of |U|'*|A| and |V|'*|B|. */ q__4.r = -d1.r, q__4.i = -d1.i; q__3.r = snr * q__4.r, q__3.i = snr * q__4.i; q__2.r = *a1 * q__3.r, q__2.i = *a1 * q__3.i; q__5.r = csr * a2->r, q__5.i = csr * a2->i; q__1.r = q__2.r + q__5.r, q__1.i = q__2.i + q__5.i; ua21.r = q__1.r, ua21.i = q__1.i; ua22r = csr * *a3; q__4.r = -d1.r, q__4.i = -d1.i; q__3.r = snl * q__4.r, q__3.i = snl * q__4.i; q__2.r = *b1 * q__3.r, q__2.i = *b1 * q__3.i; q__5.r = csl * b2->r, q__5.i = csl * b2->i; q__1.r = q__2.r + q__5.r, q__1.i = q__2.i + q__5.i; vb21.r = q__1.r, vb21.i = q__1.i; vb22r = csl * *b3; aua21 = dabs(snr) * dabs(*a1) + dabs(csr) * ((r__1 = a2->r, dabs( r__1)) + (r__2 = r_imag(a2), dabs(r__2))); avb21 = dabs(snl) * dabs(*b1) + dabs(csl) * ((r__1 = b2->r, dabs( r__1)) + (r__2 = r_imag(b2), dabs(r__2))); /* zero (2,1) elements of U'*A and V'*B. */ if ((r__1 = ua21.r, dabs(r__1)) + (r__2 = r_imag(&ua21), dabs( r__2)) + dabs(ua22r) == 0.f) { q__1.r = vb22r, q__1.i = 0.f; clartg_(&q__1, &vb21, csq, snq, &r__); } else if ((r__1 = vb21.r, dabs(r__1)) + (r__2 = r_imag(&vb21), dabs(r__2)) + dabs(vb22r) == 0.f) { q__1.r = ua22r, q__1.i = 0.f; clartg_(&q__1, &ua21, csq, snq, &r__); } else if (aua21 / ((r__1 = ua21.r, dabs(r__1)) + (r__2 = r_imag(& ua21), dabs(r__2)) + dabs(ua22r)) <= avb21 / ((r__3 = vb21.r, dabs(r__3)) + (r__4 = r_imag(&vb21), dabs(r__4)) + dabs(vb22r))) { q__1.r = ua22r, q__1.i = 0.f; clartg_(&q__1, &ua21, csq, snq, &r__); } else { q__1.r = vb22r, q__1.i = 0.f; clartg_(&q__1, &vb21, csq, snq, &r__); } *csu = csr; r_cnjg(&q__3, &d1); q__2.r = -q__3.r, q__2.i = -q__3.i; q__1.r = snr * q__2.r, q__1.i = snr * q__2.i; snu->r = q__1.r, snu->i = q__1.i; *csv = csl; r_cnjg(&q__3, &d1); q__2.r = -q__3.r, q__2.i = -q__3.i; q__1.r = snl * q__2.r, q__1.i = snl * q__2.i; snv->r = q__1.r, snv->i = q__1.i; } else { /* Compute the (1,1) and (1,2) elements of U'*A and V'*B, and (1,1) element of |U|'*|A| and |V|'*|B|. */ r__1 = csr * *a1; r_cnjg(&q__4, &d1); q__3.r = snr * q__4.r, q__3.i = snr * q__4.i; q__2.r = q__3.r * a2->r - q__3.i * a2->i, q__2.i = q__3.r * a2->i + q__3.i * a2->r; q__1.r = r__1 + q__2.r, q__1.i = q__2.i; ua11.r = q__1.r, ua11.i = q__1.i; r_cnjg(&q__3, &d1); q__2.r = snr * q__3.r, q__2.i = snr * q__3.i; q__1.r = *a3 * q__2.r, q__1.i = *a3 * q__2.i; ua12.r = q__1.r, ua12.i = q__1.i; r__1 = csl * *b1; r_cnjg(&q__4, &d1); q__3.r = snl * q__4.r, q__3.i = snl * q__4.i; q__2.r = q__3.r * b2->r - q__3.i * b2->i, q__2.i = q__3.r * b2->i + q__3.i * b2->r; q__1.r = r__1 + q__2.r, q__1.i = q__2.i; vb11.r = q__1.r, vb11.i = q__1.i; r_cnjg(&q__3, &d1); q__2.r = snl * q__3.r, q__2.i = snl * q__3.i; q__1.r = *b3 * q__2.r, q__1.i = *b3 * q__2.i; vb12.r = q__1.r, vb12.i = q__1.i; aua11 = dabs(csr) * dabs(*a1) + dabs(snr) * ((r__1 = a2->r, dabs( r__1)) + (r__2 = r_imag(a2), dabs(r__2))); avb11 = dabs(csl) * dabs(*b1) + dabs(snl) * ((r__1 = b2->r, dabs( r__1)) + (r__2 = r_imag(b2), dabs(r__2))); /* zero (1,1) elements of U'*A and V'*B, and then swap. */ if ((r__1 = ua11.r, dabs(r__1)) + (r__2 = r_imag(&ua11), dabs( r__2)) + ((r__3 = ua12.r, dabs(r__3)) + (r__4 = r_imag(& ua12), dabs(r__4))) == 0.f) { clartg_(&vb12, &vb11, csq, snq, &r__); } else if ((r__1 = vb11.r, dabs(r__1)) + (r__2 = r_imag(&vb11), dabs(r__2)) + ((r__3 = vb12.r, dabs(r__3)) + (r__4 = r_imag(&vb12), dabs(r__4))) == 0.f) { clartg_(&ua12, &ua11, csq, snq, &r__); } else if (aua11 / ((r__1 = ua11.r, dabs(r__1)) + (r__2 = r_imag(& ua11), dabs(r__2)) + ((r__3 = ua12.r, dabs(r__3)) + (r__4 = r_imag(&ua12), dabs(r__4)))) <= avb11 / ((r__5 = vb11.r, dabs(r__5)) + (r__6 = r_imag(&vb11), dabs(r__6)) + (( r__7 = vb12.r, dabs(r__7)) + (r__8 = r_imag(&vb12), dabs( r__8))))) { clartg_(&ua12, &ua11, csq, snq, &r__); } else { clartg_(&vb12, &vb11, csq, snq, &r__); } *csu = snr; r_cnjg(&q__2, &d1); q__1.r = csr * q__2.r, q__1.i = csr * q__2.i; snu->r = q__1.r, snu->i = q__1.i; *csv = snl; r_cnjg(&q__2, &d1); q__1.r = csl * q__2.r, q__1.i = csl * q__2.i; snv->r = q__1.r, snv->i = q__1.i; } } return 0; /* End of CLAGS2 */ } /* clags2_ */