Lriemannsiegel_blfi.cc 37.9 KB
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/*-------------------------------------------------------------------------------
 * VERSION: Sept. 23th, 2008.
 *------------------------------------------------------------------------------ 
 *
 * Code to compute the Riemann zeta function on the critical line using
 * band limited interpolation based ideas.
 *
 * Note: if run into memory problems, reduce blfi_block_growth
 * -----------------------------------------------------------------------------*/   

//#include<iostream>
//#include<iomanip>
//#include<cmath>
//#include<complex>

#include "L.h"
#include "Lriemannsiegel_blfi.h"

using namespace std;

//typedef complex<double> Complex;
//typedef double Double;
//typedef int int;

const int blfi=1;

//-----------------Global constants-----------------------------------------------------------------------------
//const Double Pi=3.14159265358979323846264338327950288;
//const Complex I=Complex(0.,1.);
//const Double bernoulli[] ={1.0, -1./2, 1./6, 0, -1./30, 0, 1./42, 0, -1./30, 0, 5./66, 0, -691./2730, 0, 7./6, 0, -3617./510, 0, 43867./798, 0, -174611./330, 0, 854513./138, 0, -236364091./2730, 0, 8553103./6, 0, -23749461029./870, 0, 8615841276005./14322, 0, -7709321041217./510, 0, 2577687858367./6, 0, -26315271553053477373./1919190, 0, 2929993913841559./6, 0, -261082718496449122051./13530, 0, 1520097643918070802691./1806, 0, -27833269579301024235023./690, 0, 596451111593912163277961./282, 0, -5609403368997817686249127547./46410, 0};

/* -------moved to Lglobals.cc and .h
const Double sin_cof[]={1.,-1./6.,1./120.,-1./5040.,1./362880.,-1./39916800.};
const Double sinh_mult_fac=.5;
const Double sin_tol=1.e-5;
const int sin_terms=2;

const Double blfi_block_growth=2.; // how fast blfi blocks grow as we traverse the main sum, keep as is for now
const Double beta_fac_mult=6.;  // controls the density of blfi sampling and the number of blfi terms needed
const Double blfi_fac=.085;  // efficiency of the blfi interpolation sum relative to an RS sum of same length
const Double pts_array_fac=10000;

const int rs_blfi_N=3;

//-----------------Global arrays-----------------------------------------------------------------------------
Double *klog0; //log(k) at the beginning
Double *klog2; //log(k) at the end if needed
Double *ksqrt0; // 1/sqrt(k) at the beginning
Double *ksqrt2;// 1/sqrt(k) at the end if needed
int *num_blocks; // number of blocks
int *size_blocks;// size of blocks
Double *trig; // stores correction terms
Double *zz; // store powers of fractional part

Double **klog1; //log(k) in the middle if needed
Double **ksqrt1; // 1/sqrt(k) in the middle if needed
Double **klog_blfi; //initial term
Double **qlog_blfi; //band-width
Double **piv_org; //original pivot
Double **bbeta; //beta
Double **blambda; //lambda
Double **bepsilon; //epsilon
Double **arg_blfi; //arg_blfi
Double **inv_arg_blfi; //inv_arg_blfi

Double ***qlog_blfi_dense; // log(1+k/v) terms
Double ***qsqrt_blfi_dense; // 1/sqrt(1+k/v)
int ***blfi_done_left; //block done or not
int ***blfi_done_right; //block done or not
Double ***blfi_val_re_left; //real value of block
Double ***blfi_val_re_right; //real value of block
Double ***blfi_val_im_left; //imag value of block
Double ***blfi_val_im_right; //imag value of block


//-----------------Global variables-----------------------------------------------------------------------------
int length_org=0; // length of the main sum
int length_split=0; // length of the portion of the main sum to be evaluated directly
int lgdiv=0; // number of divisions of the main sum into intervals of the form [N,2N)
int max_pts=0; // max number of interpolation points allowed
int range=0; // number of blfi interpolation points needed
int blfi_block_size_org=0; // starting length of the blfi block
int total_blocks=0;
int rs_flag=0;

Double bc=0;
Double bc2=0.;
Double kernel_fac=0.;
Double ler=0.;
Double mult_fac=0.;
Double approx_blfi_mean_spacing=0.;
Double interval_length=0.;
Double error_tolerance=0.;
Double input_mean_spacing=0.;

------------------------------ global variables moved */

//compute sinc(x)=sin(x)/x
Double sinc(Double u){
    Double ans=1;
    if(fabs(u) > sin_tol) ans=sin(u)/u;
    if(fabs(u)<= sin_tol){
        Double u2=pow(u,2),temp=u2;
        for(int j=1;j<sin_terms;j++){
            ans=ans+sin_cof[j]*temp;
            temp=u2*temp;
        }
    }

    return ans;
}

//computes the kernel function in BLFI
Double kernel(Double u){
    Double ans=1;

    if(fabs(u) > sin_tol){
        Double temp0=exp(u);
               ans=(temp0-1/temp0)/u;
    }

    if(fabs(u)<= sin_tol){
        Double u2=pow(u,2),temp=u2;
        for(int j=1;j<sin_terms;j++){
            ans=ans+fabs(sin_cof[j])*temp;
            temp=u2*temp;
        }
        ans=ans/sinh_mult_fac;
    }

    return ans;
}

//compute the exponent of the rotation factor for zeta on the critical line
Double theta_r(Double t){
        Double mt0=t/2*log(t/(2*Pi*exp((Double)1)))-Pi/8;
        Double mt=mt0/(2*Pi);

        if(mt<0){
                cout<<"Error: t=imag(s) is too small"<<"\n";
                return 0;
        }

        int it=(int)mt;
        Double ft=mt-(Double)it;
        Double p2,p3,p5,p7;
        p2=t*t;
        p3=p2*t;
        p5=p3*p2;
        p7=p5*p2;
        ft=ft+(1/(48*t)+7/(5760*p3)+31/(80640*p5)+127/(430080*p7))/(2*Pi);
        ft=ft-(int)ft;
        Double tt=ft*2*Pi;

        return tt;
}

//compute the rotation factor for zeta on the critical line
Complex theta(Double t){
        Complex val,val1;
        Double tt=theta_r(t);
        val1=Complex(0,tt);
        val=exp(-val1);

    return val;
}

//computes the remainder term in RS 
Double remain(Double t){
        Double res,z,sgn;
    Double tau=sqrt(t/(2*Pi));
    Double frac=fmod(tau,1); 
    if(frac>=1.) frac=0;

        z=frac-(Double)1/2;

    zz[0]=1;
        for(int i=1;i<51;i++){
            zz[i]=z*zz[i-1];
    }

    trig[0]=
     .3826834323650897717284599840304*zz[0]
        +1.74896187231008179744118586948533*zz[2]
        +2.11802520768549637318456427826417*zz[4]
        -.87072166705114807391892407738239*zz[6]
        -3.4733112243465167073064116693758*zz[8]
        -1.66269473089993244964313630119286*zz[10]
        +1.21673128891923213447689352804417*zz[12]
        +1.30143041610079757730060538099786*zz[14]
        +.03051102182736167242108987123981*zz[16]
        -.37558030515450952427981932122934*zz[18]
        -.10857844165640659743546975901329*zz[20]
        +.05183290299954962337576051067322*zz[22]
        +.02999948061990227592040084956912*zz[24]
        -.00227593967061256422601994851021*zz[26]
        -.00438264741658033830594007013585*zz[28]
        -.00040642301837298469930723272116*zz[30]
        +.00040060977854221139278910314608*zz[32]
        +8.97105799138884129783418195378689e-05*zz[34]
        -2.30256500272391071161029452573899e-05*zz[36]
        -9.38000660190679248471972940127474e-06*zz[38]
        +6.32351494760910750424986123959430e-07*zz[40]
        +6.55102281923150166621223123133411e-07*zz[42]
        +2.21052374555269725866086890381876e-08*zz[44]
        -3.32231617644562883503133517017624e-08*zz[46]
        -3.73491098993365608176460476015222e-09*zz[48]
        +1.24450670607977391951510000249366e-09*zz[50];
                
        trig[1]=
    -.05365020525675069405998280791133*zz[1]
    +.11027818741081482439896362071917*zz[3]
    +1.23172001543152263131956529162206*zz[5]
    +1.26349648627994578841755482191213*zz[7]
    -1.69510899755950301844944739906596*zz[9]
    -2.99987119676501008895548735894141*zz[11]
    -.10819944959899208642692257787438*zz[13]
    +1.94076629462127126879387632539716*zz[15]
    +.78384235615006865328843457488694*zz[17]
    -.50548296679003659187902141326173*zz[19]
    -.3845072349605797405134273885311*zz[21]
    +.03747264646531532067594447494023*zz[23]
    +.09092026610973176317258142450576*zz[25]
    +.01044923755006450921820113972659*zz[27]
    -.01258297965158341649747892224592*zz[29]
    -.00339950372115127408505894886137*zz[31]
    +.00104109505377148912682954240655*zz[33]
    +.00050109490511184868603556526727*zz[35]
    -3.95635966900318155954711855696337e-05*zz[37]
    -4.76245924535718963865409830268035e-05*zz[39]
    -1.85393553380851322734349064569117e-06*zz[41]
    +3.19369180800689720404663539343268e-06*zz[43]
    +4.09078076085060663265089453677018e-07*zz[45]
    -1.54466243325766321284375723273104e-07*zz[47]
    -3.46630749176913317222559405934073e-08*zz[49];

        trig[2]=
    .00518854283029316849378458151923*zz[0]
    +.00123786335522538984133826974438*zz[2]
    -.18137505725166997411491896409414*zz[4]
    +.14291492748532126541165603376514*zz[6]
    +1.33033917666875653250993329998546*zz[8]
    +.35224723534037336775327655505836*zz[10]
    -2.4210015958919507237815305433405*zz[12]
    -1.67607870225381088533346181492372*zz[14]
    +1.36894167233283721842349153807076*zz[16]
    +1.55390194302229832214563952655935*zz[18]
    -.17221642734729980519582586998918*zz[20]
    -.63590680550454309889704902355845*zz[22]
    -.09911649873041208105423564341370*zz[24]
    +.14033480067387008950738254898316*zz[26]
    +.04782352019827292236438803506512*zz[28]
    -.01735604064147978079795864709223*zz[30]
    -.01022501253402859184447660413126*zz[32]
    +.00092741491597948878994270014371*zz[34]
    +.00135721943723733853452533619958*zz[36]
    +6.41369012029388008996238736394533e-05*zz[38]
    -.00012300805698196629883342322937*zz[40]
    -1.83135074047892025547675543979621e-05*zz[42]
    +7.82162860432262730850139938461872e-06*zz[44]
    +2.00875424847599455034985293919157e-06*zz[46]
    -3.35327653931857137372749727241453e-07*zz[48]
    -1.46160209174182309264510097122760e-07*zz[50];

        trig[3]=
    -.00267943218143891380853967145989*zz[1]
    +.02995372109103514963731329491570*zz[3]
    -.04257017254182869798501935111688*zz[5]
    -.28997965779803887506893209478669*zz[7]
    +.48888319992354459725374746407169*zz[9]
    +1.23085587639574608119312504336294*zz[11]
    -.82975607085274087041796910432976*zz[13]
    -2.24976353666656686652045012659903*zz[15]
    +.07845139961005471379365473620184*zz[17]
    +1.74674928008688940039198666645219*zz[19]
    +.45968080979749935109237306173169*zz[21]
    -.66193534710397749464339040008983*zz[23]
    -.31590441036173634578979632973316*zz[25]
    +.12844792545207495988511847476209*zz[27]
    +.10073382716626152300969450207513*zz[29]
    -.00953018384882526775950465984230*zz[31]
    -.01926442168751408889840098069714*zz[33]
    -.00124646371587692917124790716458*zz[35]
    +.00242439696411030857397215245841*zz[37]
    +.00043764769774185701827561290396*zz[39]
    -.00020714032687001791275913078304*zz[41]
    -6.27434450418651556052610958029804e-05*zz[43]
    +1.15753438145956693483789208989316e-05*zz[45]
    +5.88385492454037978388597885697078e-06*zz[47]
    -3.12467740069633622086961449076033e-07*zz[49];
        
    sgn=pow(-(Double)1,length_org-1);
    
    res=0;
    for(int i=0;i<rs_blfi_N+1;i++){
            res= res+pow(tau,-i)*trig[i];
    }
    res= sgn*pow(tau,-(Double)1/2)*res;

    return res;
}


//precomputes log(k) and sqrt(k) for the initial segement of RS
void init_klog0(){
    for(int i=1;i<length_split;i++){
        klog0[i]=LOG(i);
        ksqrt0[i]=two_inverse_sqrt(i)/2;
    }
}

//computes the initial segement of RS
Double block0_r(Double t,int start, int end){
        Double temp,term,sum=0;
        Double th=theta_r(t);

        for(int j=start;j<end;j++){
                temp=cos(t*klog0[j]-th);
                term=temp*ksqrt0[j];
                sum=sum+term;
        }

    return sum;
}

//precomputes each block at evenly spaced points on demand. Precomputed values are used in blfi interpolation formula blfi_inter(). 
Double blfi_fun(int i,int j,int n,int ll,int md){
    Double res=0,sum_re=0,sum_im=0,fp=0;
    Complex sum=0;

    if(n<0){
        if(blfi_done_left[i][j][-n]==1) {
            if(md==1) res=blfi_val_re_left[i][j][-n];
            if(md==2) res=blfi_val_im_left[i][j][-n];
        }

        if(blfi_done_left[i][j][-n]==0){
            Double u=(n+piv_org[i][j])*arg_blfi[i][j];

            for(int k=0;k<ll;k++){
                fp=u*qlog_blfi_dense[i][j][k];
                sum_re=sum_re+cos(fp)*qsqrt_blfi_dense[i][j][k];
                sum_im=sum_im+sin(fp)*qsqrt_blfi_dense[i][j][k];
            }

            Complex center_fac=exp(Complex(0,-u*qlog_blfi[i][j]));
            sum=center_fac*(sum_re+I*sum_im);

            blfi_val_re_left[i][j][-n]=real(sum);
            blfi_val_im_left[i][j][-n]=imag(sum);
            blfi_done_left[i][j][-n]=1;
            if(md==1) res=blfi_val_re_left[i][j][-n];
            if(md==2) res=blfi_val_im_left[i][j][-n];
        }
    }

    if(n>=0){
        if(blfi_done_right[i][j][n]==1) {
            if(md==1) res=blfi_val_re_right[i][j][n];
            if(md==2) res=blfi_val_im_right[i][j][n];
        }

        if(blfi_done_right[i][j][n]==0){
            Double u=(n+piv_org[i][j])*arg_blfi[i][j];
        
            for(int k=0;k<ll;k++){
                fp=u*qlog_blfi_dense[i][j][k];
                sum_re=sum_re+cos(fp)*qsqrt_blfi_dense[i][j][k]; 
                sum_im=sum_im+sin(fp)*qsqrt_blfi_dense[i][j][k]; 
            }

            Complex center_fac=exp(Complex(0,-u*qlog_blfi[i][j]));
            sum=center_fac*(sum_re+I*sum_im);

            blfi_val_re_right[i][j][n]=real(sum);
            blfi_val_im_right[i][j][n]=imag(sum);    
            blfi_done_right[i][j][n]=1;
            if(md==1) res=blfi_val_re_right[i][j][n];
            if(md==2) res=blfi_val_im_right[i][j][n];
        }
    }

    return res;
}

//computes a block using the band-limited function interpolation formula. Sums are centered.
Complex blfi_inter(Double t,Double v_denom,int i,int j,int ll, int &success){
    Double temp,temp1,temp2,temp3,temp4,temp5,temp6,temp7,temp8,temp9,temp10,temp11;
    Double sum_re=0,sum_im=0;
    Complex sum=0;

    temp3=inv_arg_blfi[i][j]*t;
    Double bpivot=temp3-fmod(temp3,1);
    int shift=(int)(bpivot-piv_org[i][j]);
    Complex decenter_fac=exp(Complex(0,t*qlog_blfi[i][j]));

    temp1=(shift-range)*arg_blfi[i][j];
    temp9=piv_org[i][j]*arg_blfi[i][j];
    temp11=temp9-t;

    if(abs(shift-range+1)>max_pts || abs(shift+range-1) >max_pts){
        success=0;
        return 0;
    }

    if(blfi==1){
        for(int n=shift-range+1;n<shift+range;n++){
            temp1=temp1+arg_blfi[i][j];
            temp2=temp1+temp11;
            temp5=blambda[i][j]*temp2;
            temp6=ler*temp5;
            temp7=bc2-pow(temp6,2);

            if(temp7<0) {
                cout<<"range not optimal."<<"\n";
                return 0;
            }

            temp8=sqrt(temp7);
            temp4=sinc(temp5)*kernel(temp8);

            sum_re=sum_re+blfi_fun(i,j,n,ll,1)*temp4;
            sum_im=sum_im+blfi_fun(i,j,n,ll,2)*temp4;
        }

        sum=decenter_fac*mult_fac*(sum_re+I*sum_im);
    }

    if(blfi==0){
        for(int n=0;n<ll;n++){
            temp=log(1+(Double)n/v_denom);
            sum=sum+exp(Complex(-temp/2,t*temp));
        }
    }

    success=1;
    return sum;
}

//initializes various paramteres and arrays needed for band-limited interpolation.
void init_blfi(Double t){
    int tail_blfi,length_blfi,div_blfi,blfi_fin,blfi_remain;
    int denom;

    int blfi_begin=length_split;
    int blfi_block_size=blfi_block_size_org;

    for (int i=1;i<lgdiv+1;i++){
        tail_blfi=blfi_begin%blfi_block_size;
        length_blfi=blfi_begin-tail_blfi;
        div_blfi=length_blfi/blfi_block_size;

        for(int j=0;j<div_blfi;j++){
            denom=blfi_begin+j*blfi_block_size;
            klog_blfi[i][j]=LOG(denom);
            qlog_blfi[i][j]=log(1+(Double)blfi_block_size/denom)/2;
            bbeta[i][j]=beta_fac_mult*qlog_blfi[i][j];
            blambda[i][j]=(beta_fac_mult+1)*qlog_blfi[i][j]/2;
            bepsilon[i][j]=(beta_fac_mult-1)*qlog_blfi[i][j]/2;

            if(bbeta[i][j]<= qlog_blfi[i][j]){
                cout<<"Error: choice of beta is producing beta <= tau!"<<"\n";
                return;
            }
            
            arg_blfi[i][j]=Pi/bbeta[i][j];
            inv_arg_blfi[i][j]=bbeta[i][j]/Pi;

            Double temp=bbeta[i][j]*t/Pi;
            piv_org[i][j]=temp-fmod(temp,1);
            
            for(int k=0;k<blfi_block_size;k++){
                qlog_blfi_dense[i][j][k]=log(1+(Double)k/denom);
                qsqrt_blfi_dense[i][j][k]=1/sqrt((1+(Double)k/denom));
            }
        }

        blfi_remain=blfi_begin-length_blfi;
        
        if(blfi_remain*blfi_fac<2*range){
            for(int j=blfi_begin+length_blfi;j<2*blfi_begin;j++){
                klog1[i][j-blfi_begin-length_blfi]=LOG(j);
                ksqrt1[i][j-blfi_begin-length_blfi]=two_inverse_sqrt(j)/2;        
            }
        }

        if(blfi_remain*blfi_fac>=2*range){
            denom=blfi_begin+length_blfi;
            klog_blfi[i][div_blfi]=LOG(denom);
            qlog_blfi[i][div_blfi]=log(1+(Double)blfi_remain/denom)/2;
            bbeta[i][div_blfi]=beta_fac_mult*qlog_blfi[i][div_blfi];
            blambda[i][div_blfi]=(beta_fac_mult+1)*qlog_blfi[i][div_blfi]/2;
            bepsilon[i][div_blfi]=(beta_fac_mult-1)*qlog_blfi[i][div_blfi]/2;

            if(bbeta[i][div_blfi]<= qlog_blfi[i][div_blfi]){
                cout<<"Error: choice of beta is producing beta <= tau!"<<"\n";
                return;
            }

            arg_blfi[i][div_blfi]=Pi/bbeta[i][div_blfi];
            inv_arg_blfi[i][div_blfi]=bbeta[i][div_blfi]/Pi;

            Double temp=bbeta[i][div_blfi]*t/Pi;
            piv_org[i][div_blfi]=temp-fmod(temp,1);

            for(int k=0;k<blfi_remain;k++){
                qlog_blfi_dense[i][div_blfi][k]=log(1+(Double)k/denom);
                qsqrt_blfi_dense[i][div_blfi][k]=1/sqrt((1+(Double)k/denom));
            }
        }
            
        blfi_begin=2*blfi_begin;    
        blfi_block_size=(int)(blfi_block_growth*blfi_block_size);
    }

    blfi_fin=length_org-blfi_begin;
    tail_blfi=blfi_fin%blfi_block_size;
    length_blfi=blfi_fin-tail_blfi;
    div_blfi=length_blfi/blfi_block_size;

    for(int j=0;j<div_blfi;j++){
        denom=blfi_begin+j*blfi_block_size;
        klog_blfi[lgdiv+1][j]=LOG(denom);
        qlog_blfi[lgdiv+1][j]=log(1+(Double)blfi_block_size/denom)/2;
        bbeta[lgdiv+1][j]=beta_fac_mult*qlog_blfi[lgdiv+1][j];
        blambda[lgdiv+1][j]=(beta_fac_mult+1)*qlog_blfi[lgdiv+1][j]/2;
        bepsilon[lgdiv+1][j]=(beta_fac_mult-1)*qlog_blfi[lgdiv+1][j]/2;

        if(bbeta[lgdiv+1][j]<= qlog_blfi[lgdiv+1][j]){
            cout<<"Error: choice of beta is producing beta <= tau!"<<"\n";
            return;
        }
            
        arg_blfi[lgdiv+1][j]=Pi/bbeta[lgdiv+1][j];
        inv_arg_blfi[lgdiv+1][j]=bbeta[lgdiv+1][j]/Pi;
        
        Double temp=bbeta[lgdiv+1][j]*t/Pi;
        piv_org[lgdiv+1][j]=temp-fmod(temp,1);

        for(int k=0;k<blfi_block_size;k++){
            qlog_blfi_dense[lgdiv+1][j][k]=log(1+(Double)k/denom);
            qsqrt_blfi_dense[lgdiv+1][j][k]=1/sqrt((1+(Double)k/denom));
        }
    }

    blfi_remain=length_org+1-blfi_begin-length_blfi;

    if(blfi_remain*blfi_fac<2*range){
        for(int j=blfi_begin+length_blfi;j<length_org+1;j++){
            klog2[j-blfi_begin-length_blfi]=LOG(j);
            ksqrt2[j-blfi_begin-length_blfi]=two_inverse_sqrt(j)/2;
        }
    }

    if(blfi_remain*blfi_fac>=2*range){
        denom=blfi_begin+length_blfi;
        klog_blfi[lgdiv+1][div_blfi]=LOG(denom);
        qlog_blfi[lgdiv+1][div_blfi]=log(1+(Double)blfi_block_size/denom)/2;
        bbeta[lgdiv+1][div_blfi]=beta_fac_mult*qlog_blfi[lgdiv+1][div_blfi];
        blambda[lgdiv+1][div_blfi]=(beta_fac_mult+1)*qlog_blfi[lgdiv+1][div_blfi]/2;
        bepsilon[lgdiv+1][div_blfi]=(beta_fac_mult-1)*qlog_blfi[lgdiv+1][div_blfi]/2;

        if(bbeta[lgdiv+1][div_blfi]<= qlog_blfi[lgdiv+1][div_blfi]){
            cout<<"Error: choice of beta is producing beta <= tau !"<<"\n";
            return;
        }
        
        arg_blfi[lgdiv+1][div_blfi]=Pi/bbeta[lgdiv+1][div_blfi];
        inv_arg_blfi[lgdiv+1][div_blfi]=bbeta[lgdiv+1][div_blfi]/Pi;
        
        Double temp=bbeta[lgdiv+1][div_blfi]*t/Pi;
        piv_org[lgdiv+1][div_blfi]=temp-fmod(temp,1);
        
        for(int k=0;k<blfi_remain;k++){
            qlog_blfi_dense[lgdiv+1][div_blfi][k]=log(1+(Double)k/denom);
            qsqrt_blfi_dense[lgdiv+1][div_blfi][k]=1/sqrt((1+(Double)k/denom));
        }
    }
}

//function to dynamically allocate memory to arrays
void init_arrays(int md){
    if(md==0){
        klog0=new Double[length_split+1];
        ksqrt0=new Double[length_split+1];
    }

    int tail_length=2*(int)(pow(blfi_block_growth,lgdiv)*blfi_block_size_org);
    if(tail_length<1.){
        cout<<"Error: tail_length < 1"<<"\n";
        return;
    }

    klog2=new Double[tail_length+1];
    ksqrt2=new Double[tail_length+1];
    trig=new Double[100];
    zz=new Double[100];

    klog1=new Double*[lgdiv+2];
    ksqrt1=new Double*[lgdiv+2];
    klog_blfi=new Double*[lgdiv+2]; 
    qlog_blfi=new Double*[lgdiv+2]; 
    piv_org=new Double*[lgdiv+2]; 
    bbeta=new Double*[lgdiv+2]; 
    blambda=new Double*[lgdiv+2]; 
    bepsilon=new Double*[lgdiv+2]; 
    arg_blfi=new Double*[lgdiv+2]; 
    inv_arg_blfi=new Double*[lgdiv+2]; 

    for(int i=1;i<lgdiv+2;i++){
        klog1[i]=new Double[size_blocks[i]+1];    
        ksqrt1[i]=new Double[size_blocks[i]+1];
        klog_blfi[i]=new Double[num_blocks[i]+1]; 
        qlog_blfi[i]=new Double[num_blocks[i]+1]; 
        piv_org[i]=new Double[num_blocks[i]+1]; 
        bbeta[i]=new Double[num_blocks[i]+1]; 
        blambda[i]=new Double[num_blocks[i]+1]; 
        bepsilon[i]=new Double[num_blocks[i]+1]; 
        arg_blfi[i]=new Double[num_blocks[i]+1]; 
        inv_arg_blfi[i]=new Double[num_blocks[i]+1];
    }

    blfi_done_left=new int**[lgdiv+2]; 
    blfi_done_right=new int**[lgdiv+2]; 
    blfi_val_re_left=new Double**[lgdiv+2]; 
    blfi_val_re_right=new Double**[lgdiv+2]; 
    blfi_val_im_left=new Double**[lgdiv+2]; 
    blfi_val_im_right=new Double**[lgdiv+2]; 
    qlog_blfi_dense=new Double**[lgdiv+2];
    qsqrt_blfi_dense=new Double**[lgdiv+2];

    for(int i=1;i<lgdiv+2;i++){
        blfi_done_left[i]=new int*[num_blocks[i]+1]; 
        blfi_done_right[i]=new int*[num_blocks[i]+1]; 
        blfi_val_re_left[i]=new Double*[num_blocks[i]+1]; 
        blfi_val_re_right[i]=new Double*[num_blocks[i]+1]; 
        blfi_val_im_left[i]=new Double*[num_blocks[i]+1];
        blfi_val_im_right[i]=new Double*[num_blocks[i]+1];
        qlog_blfi_dense[i]=new Double*[num_blocks[i]+1];
        qsqrt_blfi_dense[i]=new Double*[num_blocks[i]+1];

        for(int j=0;j<num_blocks[i]+1;j++){

            blfi_done_left[i][j]=new int[max_pts+1];
            blfi_done_right[i][j]=new int[max_pts+1];
            blfi_val_re_left[i][j]=new Double[max_pts+1];
            blfi_val_re_right[i][j]=new Double[max_pts+1];
            blfi_val_im_left[i][j]=new Double[max_pts+1];
            blfi_val_im_right[i][j]=new Double[max_pts+1];
            qlog_blfi_dense[i][j]=new Double[size_blocks[i]+1];
            qsqrt_blfi_dense[i][j]=new Double[size_blocks[i]+1];
        
            for(int k=0;k<max_pts+1;k++){
                blfi_done_left[i][j][k]=0;
                blfi_done_right[i][j][k]=0;
            }
        }    
    }
}

//function to de-allocate memory for arrays.
void clean_arrays(int md){

    for(int i=1; i<lgdiv+2;i++){
        delete [] klog1[i];
        delete [] ksqrt1[i];
        delete [] klog_blfi[i]; 
        delete [] qlog_blfi[i]; 
        delete [] piv_org[i]; 
        delete [] bbeta[i]; 
        delete [] blambda[i]; 
        delete [] bepsilon[i]; 
        delete [] arg_blfi[i]; 
        delete [] inv_arg_blfi[i]; 
        
        for(int j=0;j<num_blocks[i]+1;j++){
            delete [] blfi_done_left[i][j];
            delete [] blfi_done_right[i][j];
            delete [] blfi_val_re_left[i][j];
            delete [] blfi_val_re_right[i][j];
            delete [] blfi_val_im_left[i][j];
            delete [] blfi_val_im_right[i][j];
            delete [] qlog_blfi_dense[i][j];
            delete [] qsqrt_blfi_dense[i][j];
        }

        delete [] blfi_done_left[i];
        delete [] blfi_done_right[i];
        delete [] blfi_val_re_left[i];
        delete [] blfi_val_re_right[i];
        delete [] blfi_val_im_left[i];
        delete [] blfi_val_im_right[i];
        delete [] qlog_blfi_dense[i];
        delete [] qsqrt_blfi_dense[i];
        
    }

    if(md==0){
        delete [] klog0;
        delete [] ksqrt0;
    }

    delete [] klog2;
    delete [] ksqrt2;
    delete [] trig;
    delete [] zz;
    delete [] num_blocks;
    delete [] size_blocks;
    delete [] klog1;
    delete [] ksqrt1;
    delete [] klog_blfi; 
    delete [] qlog_blfi; 
    delete [] piv_org; 
    delete [] bbeta; 
    delete [] blambda; 
    delete [] bepsilon; 
    delete [] arg_blfi; 
    delete [] inv_arg_blfi;
    delete [] blfi_done_left;
    delete [] blfi_done_right;
    delete [] blfi_val_re_left;
    delete [] blfi_val_re_right;
    delete [] blfi_val_im_left;
    delete [] blfi_val_im_right;
    delete [] qlog_blfi_dense;
    delete [] qsqrt_blfi_dense;
}

//given t, this function initializes various parameters needed in the RS.
void initialize(Double t){
    Double tau=sqrt(t/(2*Pi));
    length_org= (int)tau;
}

//function to check for possible errors
int check(){
    if(length_split<1.){
        cout<<"Error: length_split < 1"<<"\n";
        return 0;
    }
    if(lgdiv<0){
        cout<<"Error: lgdiv < 0"<<"\n";
        return 0;
    }
    if(max_pts < 1.){
        cout<<"Error: max_pts < 1"<<"\n";
        return 0;
    }
    if((blfi_block_growth<1.) || (blfi_block_growth>2.)){
        cout<<"Error: blfi_block_growth not in [1,2] !"<<"\n";
        return 0;
    }
    if(beta_fac_mult<=0.){
        cout<<"Error: current choice of beta does not work!"<<"\n";
        return 0;
    }
    if(bc<1.){
        cout<<"Error: choose bc >= 1."<<"\n";
        return 0;
    }
    if(blfi_block_size_org<0){
        cout<<"Error: blfi_block_size_org must be >= 0!"<<"\n";
        return 0;
    }
    if(sin_terms>=5){
        cout<<"Error: sin_terms is larger than expected. Please add more terms to sin_cof array!"<<"\n";
        return 0;
    }
    if(range<1.){
        cout<<"Error: range is < 1"<<"\n";
        return 0;
    }

    return 1;
}

//function to to collect statistics on various parameter used in blfi
void init_blfi_simulate(){
    int tail_blfi,length_blfi,div_blfi,blfi_fin;

    int blfi_block_size_sim=blfi_block_size_org;
    int blfi_begin_sim=length_split;

    total_blocks=0;

    for (int i=1;i<lgdiv+1;i++){
        tail_blfi=blfi_begin_sim%blfi_block_size_sim;
        length_blfi=blfi_begin_sim-tail_blfi;
        div_blfi=length_blfi/blfi_block_size_sim;
        num_blocks[i]=div_blfi;
        total_blocks=total_blocks+num_blocks[i];
        size_blocks[i]=blfi_block_size_sim;
        blfi_begin_sim=2*blfi_begin_sim;
        blfi_block_size_sim=(int)(blfi_block_growth*blfi_block_size_sim);
    }

    size_blocks[lgdiv+1]=blfi_block_size_sim;
    blfi_fin=length_org-blfi_begin_sim;
    tail_blfi=blfi_fin%blfi_block_size_sim;
    length_blfi=blfi_fin-tail_blfi;
    div_blfi=length_blfi/blfi_block_size_sim;
    num_blocks[lgdiv+1]=div_blfi;
    total_blocks=total_blocks+num_blocks[lgdiv+1];
}

//Given t, function to set up various parameters needed in RS and blfi, check for errors, and allocate/clean memory for arrays.
int initialize_all(Double t,int md){
    int length_split_prev=0;
    if(md==1) length_split_prev=length_split;

    initialize(t);
    Double length_split_fac=input_mean_spacing*((Double)4/10);

    ler=(beta_fac_mult-1)/(beta_fac_mult+1);
    bc=-log(error_tolerance/2);
    bc2=pow(bc,2);
    kernel_fac=bc/sinh(bc);
    range=(int)((beta_fac_mult/((beta_fac_mult-1)/2))*bc/Pi);
    blfi_block_size_org=(int)(2*range/blfi_fac);
    int length_split0=(int)(sqrt(length_split_fac*blfi_block_size_org*length_org*2*Pi/beta_fac_mult));
    length_split=min(length_org,length_split0-length_split0%blfi_block_size_org+blfi_block_size_org);
    lgdiv=(int)(log((Double)length_org/(Double)length_split)/log((Double)2));
    mult_fac=kernel_fac*sinh_mult_fac*((beta_fac_mult+1)/2)/beta_fac_mult;
    approx_blfi_mean_spacing=Pi/(beta_fac_mult*log(1+(Double)blfi_block_size_org/length_split)/2);
    max_pts=2*(int)((2*interval_length+1)/approx_blfi_mean_spacing+4*range+2);

    num_blocks=new int[lgdiv+2];
    size_blocks=new int[lgdiv+2];

    if(length_split<=0.){
        cout<<"Error: length_split must be positive !"<<"\n";
        return 0;
    }

    init_blfi_simulate();

    Double error_blocks=2*exp(-bc)*blfi_block_size_org*max((Double) 1,pow(blfi_block_growth/sqrt((Double)2),lgdiv))*sqrt((Double)total_blocks)/(sqrt(length_split));

    while(error_blocks>error_tolerance){
        bc=bc+1;
        bc2=pow(bc,2);
        kernel_fac=bc/sinh(bc);
        range=(int)((beta_fac_mult/((beta_fac_mult-1)/2))*bc/Pi);
        blfi_block_size_org=(int)(2*range/blfi_fac);
        int length_split0=(int)(sqrt(length_split_fac*blfi_block_size_org*length_org*2*Pi/beta_fac_mult));
        length_split=min(length_org,length_split0-length_split0%blfi_block_size_org+blfi_block_size_org);
        lgdiv=(int)(log((Double)length_org/(Double)length_split)/log((Double)2));
        mult_fac=kernel_fac*sinh_mult_fac*((beta_fac_mult+1)/2)/beta_fac_mult;
        approx_blfi_mean_spacing=Pi/(beta_fac_mult*log(1+(Double)blfi_block_size_org/length_split)/2);
        max_pts=2*(int)((2*interval_length+1)/approx_blfi_mean_spacing+4*range+2);

        init_blfi_simulate();

        error_blocks=2*exp(-bc)*blfi_block_size_org*max((Double) 1,pow(blfi_block_growth/sqrt((Double)2),lgdiv))*sqrt((Double)total_blocks)/(sqrt(length_split));
    }

    if(!check()){
        delete [] num_blocks;
        delete [] size_blocks;
               return 0;
    }
    if(length_split<=0.){
        cout<<"Error: length_split must be positive!"<<"\n";
        return 0;
    }

    init_arrays(md);

    if(md==1){
        Double *klog0_temp=new Double[length_split_prev+1];
        Double *ksqrt0_temp=new Double[length_split_prev+1];

        for(int i=1;i<length_split_prev;i++){
            klog0_temp[i]=klog0[i];
            ksqrt0_temp[i]=ksqrt0[i];
        }

        delete [] klog0;
        delete [] ksqrt0;
        klog0=new Double[length_split+1];
        ksqrt0=new Double[length_split+1];

        for(int i=1;i<min(length_split,length_split_prev);i++){
            klog0_temp[i]=klog0[i];
            ksqrt0_temp[i]=ksqrt0[i];
        }

        delete [] klog0_temp;
        delete [] ksqrt0_temp;
    }

    init_klog0();
    init_blfi(t);

    return 1;
}

//function to check for possible errors after a reinitialization of blfi
int check1(Double t, int length0){
    if(length0-length_org !=0){
        clean_arrays(1);
        return initialize_all(t,1);
    }
    else
    {
        return check();
    }
}

//function that computes zeta
Complex my_zeta(Double t,int &success){
    int length0,blfi_begin,blfi_block_size,tail_blfi,length_blfi,div_blfi,blfi_fin,blfi_remain;
    Double temp_r,thr,denom,tail_lgk,tail_sqrt,ilgk,temp0,res1_r=0,res_mid=0,res_fin=0,rmterm;
    Complex temp,itemp,iterm,dummy,th,res=0,res1=0;

    blfi_begin=length_split;
    blfi_block_size=blfi_block_size_org;
    length0=length_org;

    initialize(t);
    if(!check1(t,length0)) return 0;

    th=theta(t);
    thr=theta_r(t);
    rmterm=remain(t);

    res1_r=block0_r(t,1,length_split);
    res=th*2*res1_r;
    res1=0;
    res_mid=0;

    for (int i=1;i<lgdiv+1;i++){
        tail_blfi=blfi_begin%blfi_block_size;
        length_blfi=blfi_begin-tail_blfi;
        div_blfi=length_blfi/blfi_block_size;

        for(int j=0;j<div_blfi;j++){
            if(blfi_block_size*blfi_fac<2*range-1){
                cout<<"Warning: blfi_block_size "<<blfi_block_size<<" appears too small. Program will run slow!"<<"\n";
            }

            ilgk=klog_blfi[i][j];
            itemp=Complex(-ilgk/2,t*ilgk);
            iterm=exp(itemp);
            dummy=blfi_inter(t,blfi_begin+j*blfi_block_size,i,j,blfi_block_size,success);
            if(success==0) return 0;
            res1=res1+iterm*dummy;
        }

        blfi_remain=blfi_begin-length_blfi;

        if(blfi_remain*blfi_fac<2*range){
            for(int j=blfi_begin+length_blfi;j<2*blfi_begin;j++){
                tail_lgk=klog1[i][j-blfi_begin-length_blfi];
                tail_sqrt=ksqrt1[i][j-blfi_begin-length_blfi];
                temp_r=cos(t*tail_lgk-thr)*tail_sqrt;
                res_mid=res_mid+temp_r;
            }
        }

        if(blfi_remain*blfi_fac>=2*range){
            if(blfi_block_size*blfi_fac<2*range-1){
                cout<<"Warning: blfi_block_size "<<blfi_block_size<<" appears too small. Program will run slow!"<<"\n";
            }

            ilgk=klog_blfi[i][div_blfi];
            itemp=Complex(-ilgk/2,t*ilgk);
            iterm=exp(itemp);
            dummy=blfi_inter(t,blfi_begin+length_blfi,i,div_blfi,blfi_remain,success);
            if(success==0) return 0;
            res1=res1+iterm*dummy;
        }

        blfi_begin=2*blfi_begin;
        blfi_block_size=(int)(blfi_block_growth*blfi_block_size);
    }

    blfi_fin=length_org-blfi_begin;
    tail_blfi=blfi_fin%blfi_block_size;
    length_blfi=blfi_fin-tail_blfi;
    div_blfi=length_blfi/blfi_block_size;

    for(int j=0;j<div_blfi;j++){
        if(blfi_block_size*blfi_fac<2*range-1){
            cout<<"Warning: blfi_block_size "<<blfi_block_size<<" appears too small. Program will run slow!"<<"\n";
        }

        ilgk=klog_blfi[lgdiv+1][j];
        itemp=Complex(-ilgk/2,t*ilgk);
        iterm=exp(itemp);
        dummy=blfi_inter(t,blfi_begin+j*blfi_block_size,lgdiv+1,j,blfi_block_size,success);
        if(success==0) return 0;
        res1=res1+iterm*dummy;
    }

    res_fin=0;
    blfi_remain=length_org+1-blfi_begin-length_blfi;
        
    if(blfi_remain*blfi_fac<2*range){
        for(int j=blfi_begin+length_blfi;j<length_org+1;j++){
            tail_lgk=klog2[j-blfi_begin-length_blfi];
            tail_sqrt=ksqrt2[j-blfi_begin-length_blfi];    
            temp_r=cos(t*tail_lgk-thr)*tail_sqrt;    
            res_fin=res_fin+temp_r;
        }
    }


    if(blfi_remain*blfi_fac>=2*range){
        if(blfi_block_size*blfi_fac<2*range-1){
            cout<<"Warning: blfi_block_size "<<blfi_block_size<<" appears too small. Program will run slow!"<<"\n";
        }
                
        ilgk=klog_blfi[lgdiv+1][div_blfi];
        itemp=Complex(-ilgk/2,t*ilgk);
        iterm=exp(itemp);
        dummy=blfi_inter(t,blfi_begin+length_blfi,lgdiv+1,div_blfi,blfi_remain,success);
        if(success==0) return 0;
        res1=res1+iterm*dummy;
    }

    res=res+th*(real(2*th*res1)+2*res_mid+2*res_fin+rmterm);
    
    success=1;
    return res;
}

//function that outputs basic statistics for blfi and RS
void output_detail(Double t){
    cout<<"\n";
    cout<<"length_org="<<length_org<<"\n";
    cout<<"length_split="<<length_split<<"\n";
    cout<<"number of blfi terms is approximately "<<2*range<<"\n";
    cout<<"mean spacing of blfi sampling is about "<<setprecision(6)<<approx_blfi_mean_spacing<<"\n";
    cout<<"mean spacing of input points is about "<<setprecision(6)<<input_mean_spacing<<"\n";
    cout<<"\n";
    cout<<"Division     Num blocks        Block size"<<"\n";
    cout<<"-----------------------------------------"<<"\n";
    for(int i=1;i<lgdiv+2;i++){
        cout<<"    "<<i<<"            "<<num_blocks[i]<<"               "<<size_blocks[i]<<"\n";
    }
    cout<<"-----------------------------------------"<<"\n";

    cout<<"\n";
    cout<<"initial sum ratio= "<<setprecision(5)<<(Double)length_split/length_org<<"\n";
    cout<<"blfi sample ratio= "<<setprecision(5)<<input_mean_spacing/approx_blfi_mean_spacing<<"\n";
    
    Double tau=sqrt(t/(2*Pi));
    Double error_blocks=2*exp(-bc)*blfi_block_size_org*max((Double) 1,pow(blfi_block_growth/sqrt((Double)2),lgdiv))*sqrt((Double)total_blocks)/(sqrt(length_split));
    Double error_sin=pow(sin_tol,2*sin_terms)*fabs(sin_cof[sin_terms]);
    cout<<"\n";
    cout<<"absolute error in the Riemann-Siegel formula is about  "<<setprecision(3)<<pow(tau,-rs_blfi_N-1.5)<<"\n";
    cout<<"absolute error in the evaluation of all blocks is about "<<setprecision(3)<<error_blocks<<"\n";
    cout<<"absolute error in the evaluation of sinc, sinh is about "<<setprecision(3)<<error_sin<<"\n";
}    

//give t, error tolerancee, and density of points to be computed, this function sets up various parameters appropriately.
int set_up(Double t, Double error, Double input_mean_spacing_given){
    interval_length=max(pts_array_fac,pts_array_fac*input_mean_spacing_given);
    if(input_mean_spacing_given<=0.){
        cout<<"Error: input density <=0. !"<<"\n";
        return 0;
    }
    if(error <= 0.){
        cout<<"Error: error tolerance given <= 0. !"<<"\n";
        return 0;
    }
    error_tolerance=error; 
    input_mean_spacing=input_mean_spacing_given;

    int success= initialize_all(t,0);

    Double bl_ratio=input_mean_spacing/approx_blfi_mean_spacing;
    if(bl_ratio>0.5){
        cout<<"Error: mean seperation of input points is too large!"<<"\n";
        clean_arrays(0);
        return 0;
    }

    return success;
}

//function to compute zeta and re-initialize the algorithm if an error is enountered.
Complex rs(Double t, Double error_given, Double input_mean_spacing_given, int &success, const char *return_type){
    if(!rs_flag) {
        success=set_up(t,error_given,input_mean_spacing_given);
        if(!success) return 0;
        rs_flag=1;
    }


    Complex res=0;
    res=my_zeta(t,success);

    if(success==0) {
        clean_arrays(0);
        success=set_up(t,error_given,input_mean_spacing_given);
        if(!success) return 0;
    }

    if (!strcmp(return_type,"rotated pure"))
        return res*exp(I*(imag(log_GAMMA(((Double)1/2+I*t)/2)) - (t/2)*log(Pi)));
    else return res;

}

/*
//the main function
int main(){
    //------------User input----------------------
    Double t=pow(10.,10);
    Double input_mean_spacing_given=.01; 
    Double error_tol=1E-9;
    //--------------------------------------------

    int success;
    Complex res;
    for(int v=0;v<20000;v++){
        res=rs(t,error_tol,input_mean_spacing_given,success);
        if(!success) return 0;
        cout<<setprecision(20)<<Complex(0.5,t)<<"   "<<res;
        t=t+input_mean_spacing_given;
        cout<<"\n";
    }

    cout<<"\n";
    output_detail(t);
    clean_arrays(0);
}

*/