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数值计算方法LU分解法实验

数值计算方法LU分解法实验

数值计算方法LU分解法实验LU分解法是一种常见的数值计算方法,用于解线性方程组或求解矩阵的逆。

该方法的核心思想是将一个矩阵分解为一个下三角矩阵和一个上三角矩阵的乘积。

在本篇文章中,我们将进行关于LU分解法的实验,并探讨其性能和应用。

首先,我们需要明确LU分解法的数学原理。

假设我们有一个n阶方阵A,LU分解法的目标是找到两个矩阵L和U,使得A=LU。

其中L是一个下三角矩阵,U是一个上三角矩阵。

然后,我们可以将原始的线性方程组Ax=b转化为两个新的方程组Ly=b和Ux=y。

通过求解这两个方程组,我们可以得到原始方程组的解。

接下来,我们将通过一个具体的例子来说明LU分解法的步骤和计算过程。

假设我们有以下方程组:2x+y+z=8-3x-y+2z=-11-2x+y+2z=-3首先,我们将系数矩阵A进行LU分解。

在这个例子中,我们可以得到下三角矩阵L和上三角矩阵U:L=100-1.510-1-11U=21101.52.5001然后,我们将方程组转化为Ly=b和Ux=y的形式。

解这两个方程组,可以得到y和x的值。

最终,我们可以得到方程组的解为x=2,y=3,z=-1通过以上的实例,我们可以看到LU分解法的步骤较为繁琐,但是它的结果是准确的。

那么,接下来我们将进行一系列实验,来评估LU分解法的性能和应用。

首先,我们将进行LU分解法的准确性测试。

我们将随机生成一组方程组,并使用LU分解法求解出它们的解。

然后,我们将使用该解验证原方程组,并计算出其误差。

我们重复这个过程多次,并计算平均误差。

通过这次实验,我们可以判断LU分解法的准确性。

其次,我们将评估LU分解法的计算效率。

我们将随机生成不同规模的方程组,并使用LU分解法求解它们。

然后,我们记录下求解所需的时间,并绘制出问题规模和求解时间的关系图。

通过这个实验,我们可以了解LU分解法在不同规模问题上的计算效率,从而评估其可行性和应用范围。

此外,我们还可以将LU分解法与其他数值计算方法进行比较。

高斯列主元消去法和直接三角分解法(LU分解)

高斯列主元消去法和直接三角分解法(LU分解)

实验六高斯列主元消去法和直接三角分解法(LU分解)一、实验名称:分别用高斯列主元消去法和直接三角分解法(LU分解)求方程组的解系数矩阵:10 7 8 7 常向量:107 5 6 5 88 6 10 9 67 5 9 10 7精确解为:(-60,102,-27,16)二、试验目的:分别用高斯列主元消去法和直接三角分解法(LU分解)求方程组的解,比较二者不同的特点。

三、算法描述:2、直接三角分解法(LU分解)四、源程序:1、高斯列主元消去法#include <stdio.h>void main(){float a[4][4]={{10,7,8,7},{7,5,6,5},{8,6,10,9},{7,5,9,10}}, y[4],c[4][4],x[4],d[4],m,b; int i,n,j,f;printf("请输入右端项:\n");for(i=0;i<=3;i++)scanf("%f",&y[i]);for(n=0;n<=2;n++){ m=a[n][n]; f=n;for(i=(n+1);i<=3;i++){if(m<a[i][n]){m=a[i][n]; f=i;}}if(f!=n){for(j=0;j<=3;j++){c[n][j]=a[n][j]; a[n][j]=a[f][j];a[f][j]=c[n][j];}d[n]=y[n]; y[n]=y[f]; y[f]=d[n];}for(i=(n+1);i<=3;i++){b=-a[i][n]/a[n][n];for(j=0;j<=3;j++)a[i][j]=a[n][j]*b+a[i][j];y[i]=y[n]*b+y[i];}}x[3]=y[3]/a[3][3];x[2]=(y[2]-a[2][3]*x[3])/a[2][2];x[1]=(y[1]-a[1][3]*x[3]-a[1][2]*x[2])/a[1][1];x[0]=(y[0]-a[0][3]*x[3]-a[0][2]*x[2]-a[0][1]*x[1])/a[0][0];printf("x1的值为%f\nx2的值为%f\nx3的值为%f\nx4的值为%f\n",x[0],x[1],x[2],x[3]);}2、直接三角分解法(LU分解)#include <stdio.h>void main (){float a[4][4]={{10,7,8,7},{7,5,6,5},{8,6,10,9},{7,5,9,10}},y[4], l[4][4],x[4],u[4][4],b[4]; int i,n,j;printf("请输入右端项:");for(i=0;i<=2;i++)scanf ("%f",&b[i]);for (i=0;i<=3;i++){ u[0][i]=a[0][i];l[i][0]=a[i][0]/u[0][0];}for (n=1;n<=2;n++){for(j=0;j<=(3-n);j++){u[n][n+j]=a[n][n+j];for (i=0;i<n;i++)u[n][n+j]=u[n][n+j]-l[n+j][i]*u[i][n]; }if (n=1){l[2][1]=(a[2][1]-l[2][0]*u[0][1])/u[1][1]; l[3][1]=(a[3][1]-l[3][0]*u[0][1])/u[1][1]; } else if (n=2){l[3][2]=(float)(a[3][2]-l[3][0]*u[0][2]-l[3][1]*u[1][2])/u[2][2]; } }for (n=0;n<=3;n++){y[n]=b[n];for (i=0;i<n;i++)y[n]=y[n]-l[n][i]*y[i]; }for (n=3;n>=0;n--){x[n]=y[n];for (i=3;i>n;i--)x[n]=(x[n]-u[n][i]*x[i]);x[n]=x[n]/u[n][n]; }for(i=0;i<=3;i++)printf("x%d的的值为:%f\n",i+1,x[i]); }五、输出结果1、高斯列主元消去法(1)请输入右端项:10 8 6 7X1的值为-59.999584X1的值为101.999306X1的值为-26.999817X1的值为15.999890(2)请输入右端项:5 6 7 8X1的值为-98.999306X1的值为163.998840X1的值为-40.999695X1的值为24.999817六、对算法的理解和改进改变右端项会对结果产生明显的影响,高斯列主元消去法仅考虑依次按列选取主元素,然后按行使之变到主元位置,再进行消去运算,消元结果冲掉A,计算解X冲掉常数项b,则在计算过程中由于右端项的不同解必然不同。

DNA光致分解

DNA光致分解

4036Inorg.Chem.2010,49,4036–4045 DOI:10.1021/ic901791fPhotocytotoxic Lanthanum(III)and Gadolinium(III)Complexes of Phenanthroline Bases Showing Light-Induced DNA Cleavage ActivityAkhtar Hussain,†Debojyoti Lahiri,†Mohammed S.Ameerunisha Begum,†Sounik Saha,†Ritankar Majumdar,‡Rajan R.Dighe,‡and Akhil R.Chakravarty*,††Department of Inorganic and Physical Chemistry and‡Department of Molecular Reproduction,Development,and Genetics,Indian Institute of Science,Bangalore560012,IndiaReceived September8,2009Lanthanide complexes of formulation[La(B)2(NO3)3](1-3)and[Gd(B)2(NO3)3](4-6),where B is a N,N-donor phenanthroline base,namely,1,10-phenanthroline(phen in1,4),dipyrido[3,2-d:20,30-f]quinoxaline(dpq in2,5)and dipyrido[3,2-a:20,30-c]phenazine(dppz in3,6),have been prepared,characterized from physicochemical data,and their photoinduced DNA and protein cleavage activity studied.The photocytotoxicity of the dppz complexes3and6 has been studied using HeLa cancer cells.The complexes exhibit ligand centered bands in the UV region.The dppz complexes show the lowest energy band at380nm in N,N-dimethylformamide(DMF).The La(III)complexes are diamagnetic.The Gd(III)complexes(4-6)have magnetic moments that correspond to seven unpaired electrons.The complexes are1:1electrolytic in aqueous DMF.The dpq and dppz complexes in DMF show ligand-based reductions.The complexes display moderate binding propensity to calf thymus DNA giving binding constant values in the range of5.7Â104-5.8Â105M-1with a relative order:3,6(dppz)>2,5(dpq)>1,4(phen).The binding data suggest DNAsurface and/or groove binding nature of the complexes.The complexes do not show any hydrolytic cleavage of plasmid supercoiled pUC19DNA.The dpq and dppz complexes efficiently cleave SC DNA to its nicked circular form on exposure to UV-A light of365nm at nanomolar complex concentration.Mechanistic studies reveal the involvement of singlet oxygen(1O2)and hydroxyl radical(HO•)as the cleavage active species.The complexes show binding propensity to bovine serum albumin(BSA)protein giving K BSA values of∼105M-1.The dppz complexes3and6show BSA protein cleavage activity in UV-A light of365nm.The dppz complexes3and6exhibit significant photocytotoxicity in HeLa cells giving respective IC50values of341nM and573nM in UV-A light of365nm for an exposure time of15min (IC50>100μM in dark for both the complexes).Control experiments show significant dark and phototoxicity of the dppz base alone(IC50=413nM in light with4h incubation in dark and11.6μM in dark with24h incubation).A significant decrease in the dark toxicity of the dppz base is observed on binding to the lanthanide ions while retaining similar phototoxicity.IntroductionMetal complexes that bind and cleave DNA and proteins under physiological conditions are of current interests for their varied applications in nucleic acid and protein chemistry,namely,as models for restriction enzymes,foot-printing agents,model proteases,and therapeutic agents.1-10 DNA cleavage could proceed via hydrolytic or oxidative pathways.Hydrolytic cleavage of DNA involves hydrolysis of the phosphodiester bond generating fragments that could be rejoined making the compounds as models for restriction enzymes having utility in genomic research.Ln(III),Zn(II), and Cu(II)complexes have been extensively studied for hydrolytic cleavage of DNA because of high Lewis acidity of these metal ions.3,4Oxidative cleavage of DNA generally*To whom correspondence should be addressed.E-mail:arc@ ipc.iisc.ernet.in.Phone:þ91-80-22932533.Fax:þ91-80-23600683. 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Clark,C.R.Met.Ions Biol.Syst.2001,38,43./IC Published on Web04/09/2010r2010American Chemical SocietyArticle Inorganic Chemistry,Vol.49,No.9,20104037causes degradation of the sugar and/or base moiety thus making the process suitable for foot-printing and therapeutic applications.Transition metal complexes with their tunable coordination environments and versatile redox and spectral properties have been used for oxidative cleavage of DNA and as models for the chemotherapeutic agents,namely,iron-bleomycins(FeBLMs).11-13Besides DNA cleaving agents, the chemistry of compounds showing protein cleavage activ-ity has received considerable current interests toward design-ing complexes that could show site-specific protein cleavage or could target cellular kinases that are expressed in the malignant tumor cells causing metastasis.14-17The anti-metastasis agents are required to target the enzymes belong-ing to the protein kinase family.Half-sandwich(η6-arene)-ruthenium(II)complexes and trans-[tetrachlorobis(1H-indazole)ruthenate(III)]are known to show potent antime-tastasis properties.18,19Further,there are reports on com-plexes showing hydrolytic cleavage of proteins.20-22The complexes showing oxidative cleavage of proteins are rela-tively rare.23-25Oxidative cleavage of DNA by metal complexes can be achieved in the presence of external reagents or on photo-activation.Photocleavage of DNA is of importance for its potential phototherapeutic applications as an emerging non-invasive treatment modality of cancer.26-29The photo-drug Photofrin which is an oligomeric mixture of hemato-porphyrin species,undergoes photoactivation at633nm to generate a1(π-π*)state with subsequent conversion to a 3(π-π*)state that activates molecular oxygen to formcytotoxic singlet oxygen.The PDT agents based on metal-loenediynes are known to produce diradical intermediates upon photoactivation.30Non-porphyrinic organic dyes have been used as potent photocytotoxic agents.31-34Use of metal-based agents showing photocytotoxicity is relatively new.Turro and co-workers have reported a ruthenium(II) complex as a cisplatin analogue that binds to DNA in a similar manner as cisplatin upon photoactivation and a rhodium(II)complex that shows significant photocytotoxi-city toward Hs-27human skin cells.35There are few reports on platinum(IV),ruthenium(II),and rhodium(II)complexes showing photocytotoxicity.36-39The reports from our la-boratory have shown that iron(III)and oxovanadium(IV) complexes could be designed as3d metal-based PDT agents showing photocytotoxicity in visible light.40,41The present work stems from our interest to design and study the photoinduced DNA cleavage activity of lanthanide com-plexes.The lanthanide complexes with their varied coordina-tion geometries could be suitably designed to achieve oxidative DNA cleavage activity.The Ln(III)complexes with their redox stability are expected to show poor chemical nuclease activity in the presence of cellular thiols.The hydrolytic DNA cleavage activity of Ln(III)complexes could be controlled with suitable design of the complexes to(11)(a)Burger,R.M.Chem.Rev.1998,98,1153.(b)Wolkenberg,S.E.; 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Inorg.Chem.2007,46,4368.(b)Saha,S.;Majumdar,R.;Roy,M.;Dighe,R.R.; Chakravarty,A.R.Inorg.Chem.2009,48,2652.(41)Sasmal,P.K.;Saha,S.;Majumdar,R.;Dighe,R.R.;Chakravarty,mun.2009,1703.4038Inorganic Chemistry,Vol.49,No.9,2010Hussain et al. minimize cellular dark toxicity.While there are few literaturereports on the lanthanide complexes showing DNA bindingpropensity,the study on photoinduced DNA cleavage activ-ity of lanthanide complexes is relatively unexplored exceptsome lanthanide complexes of macrocyclic organic dyes.Forexample,the Gd(III)and Lu(III)complexes of texaphyrinshave been studied as photodynamic therapy(PDT)agentswith the lutetium(III)texaphyrin(LUTRIN)showing PDTeffect in near-IR light of732nm.42,43The major biomedicalapplications of the lanthanide complexes are directed towardmagnetic resonance imaging(MRI)of cancer.44-47TheGd(III)-based complexes,namely,[Gd(DTPA)(H2O)]2-(Magnevist)and[Gd(DOTA)(H2O)]-(Dotarem)are cur-rently used as clinical MRI agents.47It would be of immenseinterest to develop the chemistry of lanthanide complexesshowing dual functionality as MRI and PDT agents.To explore the photoinduced DNA and protein cleavageactivity of La(III)and Gd(III)complexes,we have usedphotoactive planar phenanthroline bases like dipyridoqui-noxaline(dpq)and dipyridophenazine(dppz).The dpq anddppz ligands are known to generate photoexcited3(n-π*) and/or3(π-π*)state cleaving DNA on photoirradiation with high energy UV light.48Binding of these ligands to metal ions could significantly augment their DNA photo-cleavage activity in low-energy UV-A light.Parker and co-workers have recently shown that lanthanide complexes of a nonadentate ligand having a covalently attached dpq moiety exhibit UV light-induced plasmid DNA cleavage activity.49 A mechanism based on a Ln4þtransient intermediate is proposed for the DNA cleavage reaction.We have syn-thesized and characterized lanthanide complexes[La(B)2-(NO3)3](1-3)and[Gd(B)2(NO3)3](4-6),where B is a N, N-donor phenanthroline base,namely,1,10-phenanthroline (phen in1,4),dipyrido[3,2-d:20,30-f]quinoxaline(dpq in2,5) and dipyrido[3,2-a:20,30-c]phenazine(dppz in3,6),to study their photoinduced DNA cleavage activity and the mecha-nistic pathways involved in the DNA cleavage reactions (Scheme1).We have observed the presence of dual mechan-istic pathways involving type-II and photoredox processes in the DNA photocleavage reactions.Significant results of this report include observation of efficient photoinduced DNA cleavage activity of the dpq and dppz complexes at nanomo-lar concentrations and significant reduction of the cellular dark toxicity of the dppz base on binding to the lanthanide ions.Experimental SectionMaterials and Methods.All the reagents and chemicals were procured from commercial sources(SD Fine Chemicals,India; Aldrich,U.S.A.)and used without any further purification. Solvents used were purified by standard procedures.50Super-coiled(SC)pUC19DNA(cesium chloride purified)was pur-chased from Bangalore Genie(India).Tris-(hydroxymethyl)-aminomethane-HCl(Tris-HCl)buffer solution was prepared using deionized and sonicated triple distilled water using a quartz water distillation setup.Calf thymus(CT)DNA,agarose (molecular biology grade),distamycin-A,methyl green,cata-lase,superoxide dismutase(SOD),2,2,6,6-tetramethyl-4-piper-idone(TEMP),1,4-diazabicyclo[2.2.2]octan(DABCO),ethi-dium bromide(EB),Hoechst33258,bovine serum albumin (BSA)protein,acrylamide,N,N0-methylene-bis-acrylamide, ammonium persulphate,N,N,N0,N0-tetramethylethylenediamine (TEMED),2-mercaptoethanol(MPE),glycerol,sodium dodecyl sulfate(SDS),bromophenol blue,coomassie brilliant blue R-250 were from Sigma(U.S.A.).The N,N-donor heterocyclic bases dipyrido-[3,2-d:20,30-f]-quinoxaline and dipyrido[3,2-a:20,30-c]-phenazine were prepared by literature procedures using1,10-phenanthroline-5,6-dione as a precursor reacted with ethylenedia-mine for dpq and1,2-phenylenediamine for dppz.51,52The elemental analyses were done using a Thermo Finnigan Flash EA1112CHNS analyzer.The infrared spectra were recorded on a Bruker ALPHA FT-IR spectrometer.Electro-nic spectra were recorded on a PerkinElmer Spectrum one55 spectrophotometer.Molar conductivity measurements were performed using a Control Dynamics(India)conductivity meter.Room temperature magnetic susceptibility data for the Gd(III)complexes were obtained from a George Associates Inc. Lewis-coil force magnetometer using Hg[Co(NCS)4]as a stan-dard.Experimental susceptibility data were corrected for dia-magnetic contributions.53Cyclic voltammetric measurements were made at25°C on a EG&G PAR Model253VersaStat potentiostat/galvanostat with electrochemical analysis software 270using a three electrode setup comprising a glassy carbon working,platinum wire auxiliary,and a saturated calomel reference(SCE)electrode.Tetrabutylammonium perchlorate plexes1-6and the Phenanthroline BasesUsed(42)(a)Young,S.W.;Woodburn,K.W.;Wright,M.;Mody,T.D.;Fan, Q.;Sessler,J.L.;Dow,W.C.;Miller,R.A.Photochem.Photobiol.1996,63, 892.(b)Guldi,D.M.;Mody,T.D.;Gerasimchuk,N.N.;Magda,D.;Sessler,J.L. J.Am.Chem.Soc.2000,122,8289.(43)Sessler,J.L.;Miller,R.A.Biochem.Pharmacol.2000,59,733.(44)(a)Werner,E.J.;Datta,A.;Jocher,C.J.;Raymond,K.N.Angew. Chem.,Int.Ed.2008,47,8568.(b)Datta,A.;Raymond,K.N.Acc.Chem.Res. 2009,42,938.(c)Major,J.L.;Meade,T.J.Acc.Chem.Res.2009,42,893.(45)(a)Caravan,P.Acc.Chem.Res.2009,42,851.(b)Caravan,P.Chem. Soc.Rev.2006,35,512.(c)Aime,S.;Botta,M.;Fasano,M.;Terreno,E.Chem. Soc.Rev.1998,27,19.(46)(a)Parker,D.;Dickins,R.S.;Puschmann,H.;Crossland,C.; Howard,J.A.K.Chem.Rev.2002,102,1977.(b)Bottrill,M.;Kwok,L.;Long,N.J.Chem.Soc.Rev.2006,35,557.(c)Ahrens,E.T.;Rothb€a cher,U.;Jacobs,R.E.;Fraser,S.E.Proc.Natl.Acad.Sci.U.S.A.1998,95,8443. (47)(a)Caravan,P.;Ellison,J.J.;McMurry,T.J.;Lauffer,R.B.Chem.Rev.1999,99,2293.(b)Lauffer,R.B.Chem.Rev.1987,87,901.(48)Toshima,K.;Takano,R.;Ozawa,T.;Matsumura,-mun.2002,212.(49)Frias,J.C.;Bobba,G.;Cann,M.J.;Hutchison,C.J.;Parker,D.Org.Biomol.Chem.2003,1,905.(50)Perrin,D.D.;Armarego,W.L.F.;Perrin,D.R.Purification of Laboratory Chemicals;Pergamon Press:Oxford,1980.(51)(a)Dickeson,J.E.;Summers,L.A.Aus.J.Chem.1970,23,1023.(b) Collins,J.G.;Sleeman,A.D.;Aldrich-Wright,J.R.;Greguric,I.;Hambley,T.W. Inorg.Chem.1998,37,3133.(52)Amouyal,E.;Homsi,A.;Chambron,J.-C.;Sauvage,J.-P.J.Chem. Soc.,Dalton Trans.1990,1841.(53)Kahn,O.Molecular Magnetism;VCH:Weinheim,1993.Article Inorganic Chemistry,Vol.49,No.9,20104039(TBAP,0.1M)was used as a supporting electrolyte in DMF.The electrochemical data were uncorrected for junction poten-tials.Electrospray ionization mass spectral measurements were done using Bruker Daltonics make(Esquire300Plus ESIModel).Mass spectral measurements of BSA samples weredone using Bruker Daltonics Ultraflex MALDI-TOF instru-ment.1H NMR spectra of the ligands and the La(III)complexeswere recorded at room temperature on a Bruker400MHz NMRspectrometer.Synthesis of[La(B)2(NO3)3](1-3)and[Gd(B)2(NO3)3](4-6)[B=phen(in1,4);dpq(in2,5),dppz(in3,6)].The complexes 1-6were prepared by following a reported synthetic procedurein modified form in which a hot ethanolic solution of Ln(NO3)33 6H2O(Ln=La(III),0.433g;Gd(III),0.451g;1.0mmol)inboiling ethanol(30mL)was added dropwise to a stirred solution of the respective heterocyclic base B(0.40g phen;0.47g dpq;0.57g dppz;2.0mmol)in boiling ethanol(30mL).54Afterstirring for5min,a crystalline precipitate was obtained.Thesolid was isolated,washed with20mL of hot ethanol followed by20mL of diethyl ether,and finally dried in vacuum over P4O10[Yield:∼70%].The characterization data for the com-plexes are given below.[La(phen)2(NO3)3](1).Anal.Calcd for C24H18N7O9La:C,42.06;H,2.35;N,14.30.Found:C,42.32;H,2.45;N,14.16.ESI-MS in10%aqueous MeOH:m/z623[M-NO3-]þ.IR data/ cm-1:1620w,1591w,1470vs,1408s,1284s,1150w,1102w, 1027s,862w,843s,816m,763w,717vs,637m,417m(vs,very strong;s,strong;m,medium;w,weak).UV-visible in DMF [λmax/nm(ε/M-1cm-1)]:322sh(2420),310sh(4340),266(66 420)(sh,shoulder).1H NMR in DMSO-d6(δ,ppm):9.01(dd, 2H,J=8.0,1.7Hz),8.4(dd,2H,J=4.8,1.6Hz),7.9(s,2H), 7.68(dd,2H,J=8.2,4.4Hz).[La(dpq)2(NO3)3](2).Anal.Calcd for C28H18N11O9La:C,42.60;H,2.04;N,19.52.Found:C,42.86;H,2.26;N,19.31.ESI-MS in1:1MeOH-MeCN mixture:m/z727[M-NO3-]þ.IRdata/cm-1:1634w,1576m,1460vs,1395s,1287vs,1210w, 1120w,1084m,1037m,867w,813s,735vs,696w,634w,434w, 416m.UV-visible in DMF[λmax/nm(ε/M-1cm-1)]:340 (17700),325sh(21700),265(72400).1H NMR in DMSO-d6 (δ,ppm):9.37(dd,2H,J=8.3,1.8Hz),9.15(dd,2H,J=4.4, 1.8Hz),9.09(s,2H),7.88(dd,2H,J=8.3,4.2Hz).[La(dppz)2(NO3)3](3).Anal.Calcd for C36H20N11O9La:C,48.61;H,2.27;N,17.32.Found:C,48.80;H,2.29;N,17.42.ESI-MS in1:1MeOH-MeCN mixture:m/z827[M-NO3-]þ.IRdata/cm-1:1577m,1477vs,1410s,1361m,1287vs,1135w, 1080s,1025m,818s,762s,730vs,703m,637w,615w,575w, 415s.UV-visible in DMF[λmax/nm(ε/M-1cm-1)]:380(19 460),369(15900),362(18150),351(13000),294sh(30000),268 (72400).1H NMR in DMSO-d6(δ,ppm):9.44(dd,2H,J=8.2, 1.8Hz),9.11(dd,2H,J=4.5,2.0Hz),8.29-8.33(m,2H), 7.95-7.99(m,2H),7.85(dd,2H,J=8.4,4.0Hz).[Gd(phen)2(NO3)3](4).Anal.Calcd for C24H18N7O9Gd:C,40.96;H,2.29;N,13.93.Found:C,41.04;H,2.00;N,14.21.ESI-MS in10%aqueous MeOH:m/z642[M-NO3-]þ.IR data/ cm-1:1621w,1590w,1468vs,1410s,1290vs,1143m,1099m, 1027m,855s,812m,765w,717vs,636m,417m.UV-visible in DMF[λmax/nm(ε/M-1cm-1)]:322sh(3230),310sh(4850), 266(68040).μeff=7.94μB at298K.[Gd(dpq)2(NO3)3](5).Anal.Calcd for C28H18N11O9Gd:C,41.63;H,2.00;N,19.07.Found:C,41.71;H,2.15;N,19.20.ESI-MS in1:1MeOH-MeCN mixture:m/z746[M-NO3-]þ.IRdata/cm-1:1580m,1463vs,1400s,1296vs,1215w,1125w,1080 m,1027m,880w,809m,728vs,700m,637w,430m,419m. UV-visible in DMF[λmax/nm(ε/M-1cm-1)]:340(17930), 325sh(22110),265(74100).μeff=7.98μB at298K.[Gd(dppz)2(NO3)3](6).Anal.Calcd for C36H20N11O9Gd:C,47.63;H,2.22;N,16.97.Found:C,47.84;H,2.20;N,16.77.ESI-MS in1:1MeOH-MeCN mixture:m/z846[M-NO3-]þ.IR data/cm-1:1570w,1480vs,1416m,1362m,1295vs,1136w, 1073m,1028s,821s,761s,730vs,703s,641w,613w,572w,415s.UV-visible in DMF[λmax/nm(ε/M-1cm-1)]:380(20270), 369(17260),361(19450),351(14600),293sh(31000),269 (78600).μeff=7.96μB at298K.Solubility and Stability.All the complexes were soluble in DMF and DMSO.The phenanthroline complexes were solublein H2O and MeCN.The complexes showed less solubility in MeOH and EtOH.The complexes were stable in the solid state. The complexes showed decomposition on prolonged storage inthe solution phase.54The solution stability of the dppz complex 3(logβ2=10.4)was measured in10%aqueous DMF following a literature method(vide Supporting Information for details).55DNA Binding Methods.DNA binding experiments were done in Tris-HCl/NaCl buffer(5mM Tris-HCl,5mM NaCl,pH7.2)using DMF solution of the complexes1-6.Calf thymus(CT) DNA(ca.350μM NP)in this buffer medium gave a ratio of UV absorbance at260and280nm of about1.9:1indicating that the DNA is apparently free from protein.The concentration of CTDNA was estimated from its absorption intensity at260nm with a known molar extinction coefficient value(ε)of6600M-1 cm-1.56Absorption titration experiments were made by varying the concentration of the CT DNA while keeping the metalcomplex concentration constant.Due correction was made for the absorbance of CT DNA itself.Each spectrum was recorded after equilibration of the sample for5min.The intrinsicequilibrium binding constant(K b)and the binding site size(s) of the complexes1-6to CT DNA were obtained by the McGhee-von Hippel(MvH)method using the expression of Bard et al.by monitoring the change of the absorption intensityof the spectral bands with increasing concentration of CT DNA by regression analysis using the equation(εa-εf)/(εb-εf)= (b-(b2-2K b2C t[DNA]t/s)1/2)/2K b C t,where b=1þK b C tþK b[DNA]t/2s andεa is the extinction coefficient observed forthe absorption band at a given DNA concentration,εf is the extinction coefficient of the complex free in solution,εb is the extinction coefficient of the complex when fully bound to DNA,K b is the equilibrium binding constant,C t is the total metal complex concentration,[DNA]t is the DNA concentration in nucleotides,and s is the binding site size in base pairs.57,58The non-linear least-squares analyses were done using Origin Lab, version6.1.The apparent DNA binding constant(K app)values of the complexes1-6were obtained from fluorescence spectral measurements using ethidium bromide bound CT DNA solu-tion in Tris-HCl/NaCl buffer(pH,7.2).The fluorescence in-tensities of ethidium bromide at600nm(546nm excitation) were recorded with an increasing amount of the added complex concentration.Ethidium bromide showed no apparent emission in Tris-buffer medium because of fluorescence quenching of the free ethidium bromide by the solvent molecules.59In the presence of CT DNA,ethidium bromide showed significantly enhanced emission intensity.The K app values were obtained from the equa-tion:K appÂ[complex]50=K EBÂ[EB],where K app is the apparent binding constant of the complex studied,[complex]50is the concentration of the complex at50%quenching of DNA-bound ethidium bromide emission intensity,K EB is the binding constant of ethidium bromide(K EB=1.0Â107M-1),and[EB]is the concentration of ethidium bromide(1.3μM).60(54)Hart,F.A.;Laming,F.P.J.Inorg.Nucl.Chem.1965,27,1605.(55)Vallee,B.L.;Coombs,T.L.J.Biol.Chem.1959,234,2615.(56)Reichmann,M.E.;Rice,S.A.;Thomas,C.A.;Doty,P.J.Am. Chem.Soc.1954,76,3047.(57)McGhee,J.D.;von Hippel,P.H.J.Mol.Biol.1974,86,469.(58)Carter,M.T.;Rodriguez,M.;Bard,A.J.J.Am.Chem.Soc.1989, 111,8901.(59)(a)Waring,M.J.J.Mol.Biol.1965,13,269.(b)LePecq,J.-B.;Paoletti,C.J.Mol.Biol.1967,27,87.(60)Lee,M.;Rhodes,A.L.;Wyatt,M.D.;Forrow,S.;Hartley,J.A. Biochemistry1993,32,4237.4040Inorganic Chemistry,Vol.49,No.9,2010Hussain et al.DNA melting experiments were carried out by monitoring the absorbance of CT DNA(200μM)at260nm at various temperatures,both in the absence and presence of the complexes (20μM).Measurements were carried out using a Cary300bio UV-visible spectrometer with a Cary temperature controller at an increase rate of0.5°C per min of the solution.Viscometric titrations were performed with a Schott Gerate AVS310Auto-mated Viscometer that was thermostatted at37°C in a constant temperature bath.The concentration of CT DNA was150μM in NP(nucleotide pair),and the flow times were measured using an automated timer.Each sample was measured3times,and an average flow time was calculated.Data were presented as (η/η0)1/3versus[complex]/[DNA],whereηis the viscosity of DNA in the presence of complex andη0is that of DNA alone. Viscosity values were calculated from the observed flow time of DNA-containing solutions(t)corrected for that of the buffer alone(t0),η=(t-t0)/t0.Due corrections were made for the viscosity of DMF solvent present in the solution.DNA Cleavage Experiments.The cleavage of supercoiled (SC)pUC19DNA(30μM,0.2μg,2686base-pairs)was studied by agarose gel electrophoresis.For photoinduced DNA clea-vage studies,the reactions were carried out under illuminated conditions using UV-A light of365nm(6W,Model LF-206.LS of Bangalore Genei).Eppendorf vials were used for photoclea-vage experiments in a dark room at25°C using SC DNA(1μL, 30μM)in50mM Tris-HCl buffer(pH7.2)containing50mM NaCl and the complex(2μL)with varied concentrations.The concentration of the complexes in DMF or the additives in buffer corresponded to the quantity in2μL stock solution after dilution to the20μL final volume using Tris-HCl buffer.The solution path length in the sample vial was∼5mm.After light exposure,each sample was incubated for1.0h at37°C and analyzed for the photocleaved products using gel electrophore-sis.Mechanistic studies were carried out using different addi-tives(NaN3,0.5mM;TEMP,0.5mM;DABCO,0.5mM; DMSO,4μL;KI,0.5mM;catalase,4units;SOD,4units)prior to the addition of the complex.For the D2O experiment,this solvent was used for dilution of the sample to20μL final volume.The samples after incubation in a dark chamber were added to the loading buffer containing0.25%bromophenol blue,0.25%xylene cyanol,30%glycerol(3μL),and the solution was finally loaded on1%agarose gel containing1.0μg/mL ethidium bromide.Electrophoresis was carried out in a dark room for2.0h at60V in TAE(Tris-acetate EDTA)buffer. Bands were visualized in UV light and photographed.The extent of SC DNA cleavage was measured from the intensities of the bands using the UVITEC Gel Documentation System. Due corrections were made for the low level of nicked circular (NC)form of DNA present in the original SC DNA sample and for the low affinity of ethidium bromide binding to SC compared to NC and linear forms of DNA.61The observed error in measuring the band intensities was in the range3-5%.BSA Interaction and Cleavage Experiments.The protein interaction study was performed by tryptophan fluorescence quenching experiments using BSA(2μM)in phosphate buffer (pH6.8)containing15%DMF.Quenching of the emission intensity of tryptophan residues of BSA at344nm(excitation wavelength at295nm)was monitored using complexes1-6as quenchers with increasing complex concentration.62Stern-Volmer I0/I versus[complex]plots were made using the corrected fluorescence data taking into account the effect of dilution.A linear fit of the data using the equation:I0/I= 1þK BSA[Q],where I0and I are the respective emission intensities of BSA in absence and presence of the quencher of concentration[Q],gave the interaction constant(K BSA)values using Origin6.1.Photoinduced BSA cleavage experiments were carried out following literature procedure.63Freshly prepared solutions of BSA in50mM Tris-HCl buffer(pH7.2)containing0.6%DMF were used for the photochemical protein cleavage studies.The BSA solutions in Tris-HCl buffer medium containing complexes 1-6(10μM and15μM)and BSA(5μM)were photoirradiated at365nm(100W)in Eppendorf vials.The BSA solutions containing the complexes were incubated at37°C for1.0h prior to the photoexposure.The photoirradiated samples (50μL)were dried in a centrifugal vaporizer(EYELA Centri-fugal Vaporizer,Model CVE-200D),and the samples were dissolved in the loading buffer(24μL)containing SDS(7%w/v), glycerol(4%w/v),Tris-HCl buffer(50mM,pH6.8),mercap-toethanol(2%v/v),and bromophenol blue(0.01%w/v).The protein solutions were then denatured by heating on a boiling water bath for3min.The samples were loaded on a3% polyacrylamide stacking gel.The gel electrophoresis was done at an initial applied voltage of60V until the dye passed into the separating gel from the stacking(3%)gel,and then the voltage was increased to110V.The gels were run for1.5h,stained with Coomassie Brilliant Blue R-250solution(acetic acid/ methanol/water=1:2:7v/v)and destained with water/ methanol/acetic acid mixture(5:4:1v/v)for4h.The gels,after destaining,were scanned with a HP Scanjet G3010scanner,and the images were further processed using the Adobe Photoshop 7.0software package.Molecular weight markers were used in each gel to calibrate the molecular weight of the protein.The presence of reactive oxygen species was investigated by carrying out the photoinduced protein cleavage experiments using singlet oxygen quenchers such as NaN3(3mM)and TEMP (3mM)and hydroxyl radical scavengers,namely,DMSO (20μL)and KI(3mM).Cell Cytotoxicity Assay.The photocytotoxicity of the dppz complexes(3,6)and the dppz ligand was studied using3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide(MTT) assay which is based on the ability of mitochondrial dehydro-genases of viable cells to cleave the tetrazolium rings of MTT forming dark purple membrane impermeable crystals of for-mazan that could be quantified at595nm after solubilization in detergent.64Approximately,8000cells of human cervical carci-noma(HeLa)were plated in96wells culture plate in Dulbecco’s Modified Eagle Medium(DMEM)containing10%FBS,and after24h of incubation at37°C in a CO2incubator,various concentrations of the complexes3,6or dppz ligand dissolved in 1%DMSO were added to the cells and incubation was con-tinued for4h in dark.The medium was subsequently replaced with PBS and photoirradiated for15min using UV-A light of 365nm.After photoexposure,PBS was removed and replaced with DMEM-FBS,and incubation was continued for further 24h in dark.At the end of the incubation period,20μL of5mg mL-1MTT was added to each well and incubated for an additional3h.The culture medium was finally discarded,and 100μL of10%SDS/0.01M HCl was added.The plates were then incubated at37°C for6h to dissolve the formazan crystals, and the absorbance at595nm was determined using a BIORAD ELISA plate reader.Cytotoxicity of the dppz ligand and the complexes3and6was measured as the percentage ratio of the absorbance of the treated cells to the untreated controls.The IC50values were determined by non-linear regression analysis using the GraphPad Prism software.To determine the dark cytotoxicity of the complexes3and6and the dppz ligand, various concentrations of the complexes or dppz ligand dis-solved in DMSO(1%)were added to the HeLa cells and incubated for24h in dark,thereafter the media were discarded(61)Bernadou,J.;Pratviel,G.;Bennis,F.;Girardet,M.;Meunier,B.Biochemistry1989,28,7268.(62)Quiming,N.S.;Vergel,R.B.;Nicolas,M.G.;Villanueva,J.A. J.Health Sci.2005,51,8.(63)Kumar,C.V.;Buranaprapuk,A.;Sze,H.C.;Jockusch,S.;Turro, N.J.Proc.Natl.Acad.Sci.U.S.A.2002,99,5810.(64)Mosmann,T.J.Immunol.Methods1983,65,55.。

实验三 高斯消去法和三角分解法1

实验三 高斯消去法和三角分解法1

实验报告实验三 高斯消去法与矩阵的三角分解一、实验目的1、掌握列主元素消去法,并且能够用MATLAB 编写相关程序,实现高斯消去法的求解。

2、能够用矩阵理论理解与研究高斯消去法,通过对矩阵的初等变换实现高斯消去法。

3、学会矩阵的三角分解,并且能够用MATLAB 编写相关程序,实现矩阵的三角分解,解方程组。

二、上机内容⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡2822171310871234567112345611123451111234111112311111121111111764321x x x x x x1、用列主元素高斯消去法求解方程组。

2、用列主元消去法求解方程组(实现PA=LU) 要求输出: (1)计算解X;(2)L,U;(3)正整型数组IP(i),(i=1,···,n) (记录主行信息)。

三、实验原理1、列主元素消去法用高斯消去法求解方程组时,为了减小误差,在消去的过程中要避免用绝对值较小的主元素。

因此在高斯消去法的每一步应该在系数矩阵货消去后的低阶矩阵中选取绝对值较大的元素作为主元素,保持|m ik |<=1,以减小计算过程中的舍入误差对计算解的影响。

此方法为完全主元素消去法。

完全主元素消去法在选主元素时花费一定的计算机时间,因此实际计算中常用列主元消去法。

列主元消去法在每次选主元时,仅依次按列选取绝对值最大的元素作为主元素,且仅交换两行,再进行消元计算。

装订 线第k步计算如下:对于k=1,2,…,n-1(1)按列选主元:即确定t使(2)如果t≠k,则交换[A,b]第t行与第k行元素。

(3)消元计算(4)回代求解计算流程图回代求解 b=b/a (当a nn ≠0)b ←(b -∑a x )/adet=a nn *det输出计算解及行列式及detk=1,2,…,n-1输入n ,A,b,εdet=1按列主元|a i(k),k |=max|a ik |C 0=a i(k),k换行 a ik a i(k)j(j=k,…n ) b k b j(k), 消元计算 (i=k+1,…,n ) a ik=a ik -a kk *m ik a ij=a ij -a kj *m ik (j=k+1,…,n )|C 0|<εi k =kdet=a kk det否否是是k<=n-1输出det(A)=0停机停机2. 矩阵的三角分解法 (1)定理设 n n R A ⨯∈ 。

运筹学实验报告(F-R共轭梯度法、Wolfe简约梯度法)

运筹学实验报告(F-R共轭梯度法、Wolfe简约梯度法)

f ( X ( k 1) ) min f ( X ( k ) d )
0
(k ) ( k 1) (k ) 0 的解为 如此下去, 得到序列{ X }。不难求得 d , Ad
(k )
X ( k 1) X ( k )
(k )
b AX ( k ) , d ( k ) ( k ) d d ( k 1) , Ad ( k 1)
end
T(j)= new_direction(j) else T(j)=0; end if (T(j)<0) TT=abs([TT,(x_1(j)/T(j))]) end j=j+1;
n=size(TT); for uk=1:n if(tmax>TT(uk)) tmax=TT(uk) n=n+1; end end x=x_1+t*new_direction; xx=[x(1),x(2)]; f_step=subs(f,findsym(f),xx); F=diff(f_step,t); solve(F,t); t0=tmax; x_2=x_1+0.18*new_direction A=[A;x_2]; norm0=norm; search_direction=new_direction x_1=x_2; xx2=[x_2(1),x_2(1)]; Y=subs(f,findsym(f),xx2); HSZ=[HSZ;Y] k=k+1; end k x_2 norm
注意到 d 的选取不唯一,我们可取
d ( k ) f ( X ( k ) ) k 1d ( k 1)
(k ) ( k 1) 0 可得: 由共轭的定义 d , Ad
k 1

06第17讲-过程合成1解析

06第17讲-过程合成1解析
得: N min N h Nc 最简单的换热网络---投资最小
可能的组合方案数为:
N max
N max
CPN P!/ N!/(P!N!)
N m in
N m in
其中,总费用最小的 方案即为最佳方案
(4)反应器网络的合成
任务:给定动力学和进料条件下寻求适宜的反应器类型、流程 结构和关键设计参数,以使特定的目标函数最优。
组分数N
方法数P
序列数Z N
P
Z
3
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1430
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28Leabharlann 95558593750
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1
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(3) 换热网络
换热过程是化工生产中的基本过程
化工生产过程中换热网络的作用和意义
CO+3H2
30℃
14
H2
CH4+H2O 高压蒸汽
甲脱烷甲化烷塔塔
B10 B12
第6章 系统综合 (系统合成,Process system synthesis )
0概述
化工系统工程主要研究内容
系统分析 系统优化 系统合成
核心是 优化
化工过程系统工程研究的四个基本问题
操作型问题
设计型问题 操作优化 过程合成
系统分析的任务在于描述或预测系统的 特性;系统优化是为了以最佳的方式实 现系统的目标或功能;系统综合?
分离过程合成中的一些经验规则
举例:
●在所有的分离方法中,优先采用能量分离剂(精馏),避免采用质量 分离剂(如萃取);当关键组分相对挥发度小于1.1时,可采用后者。 ●首先除去腐蚀性、有毒有害的产品; ●最后处理难分离或分离要求高的的产品; ●进料中最多的组分应当首先分离出去;

matlab牛顿拉夫逊法与快速分解法的实现

matlab牛顿拉夫逊法与快速分解法的实现

一、概述MATLAB是一种强大的数学软件工具,它提供了许多优秀的数值计算和数据分析功能。

其中,牛顿拉夫逊法和快速分解法是两种常用的数值计算方法,它们在解决非线性方程组和矩阵分解等问题中具有重要的应用价值。

本文将介绍如何在MATLAB中实现这两种方法,并对它们的优缺点进行详细分析。

二、牛顿拉夫逊法的实现1. 算法原理牛顿拉夫逊法是一种用于求解非线性方程组的迭代算法。

它利用函数的一阶和二阶导数信息来不断逼近方程组的解,直到满足精度要求为止。

算法原理可以用以下公式表示:公式1其中,x表示解向量,F(x)表示方程组的函数向量,J(x)表示方程组的雅可比矩阵,δx表示解的更新量。

通过不断迭代更新x,最终得到方程组的解。

2. MATLAB代码实现在MATLAB中,可以通过编写函数来实现牛顿拉夫逊法。

以下是一个简单的示例代码:在这段代码中,首先定义了方程组的函数向量和雅可比矩阵,然后利用牛顿拉夫逊法进行迭代更新,直到满足精度要求为止。

通过这种方式,就可以在MATLAB中实现牛顿拉夫逊法,并应用于各种实际问题。

三、快速分解法的实现1. 算法原理快速分解法是一种用于矩阵分解的高效算法。

它利用矩阵的特定性质,通过分解为更小的子问题来加速计算过程。

算法原理可以用以下公式表示:公式2其中,A表示要分解的矩阵,L和U分别表示矩阵的下三角和上三角分解。

通过这种分解方式,可以将原始矩阵的计算量大大减小,提高求解效率。

2. MATLAB代码实现在MATLAB中,可以利用内置函数来实现快速分解法。

以下是一个简单的示例代码:在这段代码中,利用MATLAB内置的lu函数进行LU分解,得到矩阵的下三角和上三角分解。

通过这种方式,就可以在MATLAB中实现快速分解法,并应用于各种矩阵计算问题。

四、方法比较与分析1. 算法复杂度牛顿拉夫逊法和快速分解法在计算复杂度上有所不同。

牛顿拉夫逊法的迭代次数取决于所求解问题的非线性程度,通常需要较多的迭代次数。

用矩阵的lu三角分解法编程求解方程组

用矩阵的lu三角分解法编程求解方程组

用矩阵的lu三角分解法编程求解方程组下载提示:该文档是本店铺精心编制而成的,希望大家下载后,能够帮助大家解决实际问题。

文档下载后可定制修改,请根据实际需要进行调整和使用,谢谢!本店铺为大家提供各种类型的实用资料,如教育随笔、日记赏析、句子摘抄、古诗大全、经典美文、话题作文、工作总结、词语解析、文案摘录、其他资料等等,想了解不同资料格式和写法,敬请关注!Download tips: This document is carefully compiled by this editor. I hope that after you download it, it can help you solve practical problems. The document can be customized and modified after downloading, please adjust and use it according to actual needs, thank you! In addition, this shop provides you with various types of practical materials, such as educational essays, diary appreciation, sentence excerpts, ancient poems, classic articles, topic composition, work summary, word parsing, copy excerpts, other materials and so on, want to know different data formats and writing methods, please pay attention!用矩阵的LU三角分解法编程求解方程组在数值计算中,求解线性方程组是一个基础且常见的问题。

中南大学离散数学实验报告(实验3ABC)

中南大学离散数学实验报告(实验3ABC)

“离散数学”实验报告(实验3ABC)专业班级学号姓名日期: 2011.12.19目录一、实验目的 (3)二、实验内容 (3)三、实验环境 (3)四、实验原理和实现过程(算法描述) (3)1实验原理 (3)2实验过程 (5)五、实验数据及结果分析 (6)六、源程序清单 (10)七、其他收获及体会 (16)一、实验目的理解图论的基本概念, 图的矩阵表示, 图的连通性, 图的遍历, 以及求图的连通支方法。

二、实验内容以偶对的形式输入一个无向简单图的边, 建立该图的邻接矩阵, 判断图是否连通(A)。

并计算任意两个结点间的距离(B)。

对不连通的图输出其各个连通支(C)。

三、实验环境C或C++语言编程环境实现。

四、实验原理和实现过程(算法描述)1.实验原理(1)建立图的邻接矩阵, 判断图是否连通根据图的矩阵表示法建立邻接矩阵A, 并利用矩阵的乘法和加法求出可达矩阵, 从而判断图的连通性。

连通图的定义: 在一个无向图G 中, 若从顶点vi到顶点vj有路径相连(当然从vj到vi也一定有路径), 则称vi和vj是连通的。

如果G 是有向图, 那么连接vi 和vj的路径中所有的边都必须同向。

如果图中任意两点都是连通的, 那么图被称作连通图。

判断连通图的实现:在图中, 从任意点出发在剩余的点中, 找到所有相邻点循环, 直到没有点可以加入为止, 如果有剩余的点就是不连通的, 否则就是连通的。

或者也可用WallShell算法, 由图的邻接矩阵判断图是否连通。

(2)计算任意两个结点间的距离图中两点i, j间的距离通过检验Al中使得aij为1的最小的l值求出。

路径P中所含边的条数称为路径P的长度。

在图G<V,E>中, 从结点Vi到Vj最短路径的长度叫从Vi到Vj的距离, 记为d<Vi, Vj>。

设图的邻接矩阵是A, 则所对应的aij的值表示, 点Vi到点Vj距离为n的路径有aij条。

若aij(1), aij(2), …, aij(n-1), 中至少有一个不为0, 则可断定Vi与Vj可达, 使aij(l)≠0的最小的l即为d(Vi, Vj)。

强wolfe非线性搜索+dfp实验报告

强wolfe非线性搜索+dfp实验报告

数学与计算科学学院实验报告
实验项目名称强wolfe非精确搜索+DFP
所属课程名称最优化
实验类型算法编程
实验日期2015年12月11日
班级信计1302
学号201353100230
姓名谢刘
成绩
附录1:源程序
附录2:实验报告填写说明
1.实验项目名称:要求与实验教学大纲一致。

2.实验目的:目的要明确,要抓住重点,符合实验教学大纲要求。

3.实验原理:简要说明本实验项目所涉及的理论知识。

4.实验环境:实验用的软、硬件环境。

5.实验方案(思路、步骤和方法等):这是实验报告极其重要的内容。

概括整个实验过程。

对于验证性实验,要写明依据何种原理、操作方法进行实验,要写明需要经过哪几个步骤来实现其操作。

对于设计性和综合性实验,在上述内容基础上还应该画出流程图、设计思路和设计方法,再配以相应的文字说明。

对于创新性实验,还应注明其创新点、特色。

6.实验过程(实验中涉及的记录、数据、分析):写明具体实验方案的具体实施步骤,包括实验过程中的记录、数据和相应的分析。

7.实验结论(结果):根据实验过程中得到的结果,做出结论。

8.实验小结:本次实验心得体会、思考和建议。

9.指导教师评语及成绩:指导教师依据学生的实际报告内容,给出本次实验报告的评价。

分离工程 II --07 多级分离计算一三对角矩阵法75页PPT

分离工程 II --07 多级分离计算一三对角矩阵法75页PPT

60、生活的道路一旦选定,就要勇敢地 走到底 ,决不 回头。 ——左
56、书不仅是生活,而且是现在、过 去和未 来文化 生活的 源泉。 ——库 法耶夫 57、生命不可能有两次,但许多人连一 次也不 善于度 过。— —吕凯 特 58、问渠哪得清如许,为有源头活水来 。—— 朱熹 59、我的努力求学没有得到别的好处, 只不过 是愈来 愈发觉 自己的 角矩阵法
46、法律有权打破平静。——马·格林 47、在一千磅法律里,没有一盎司仁 爱。— —英国
48、法律一多,公正就少。——托·富 勒 49、犯罪总是以惩罚相补偿;只有处 罚才能 使犯罪 得到偿 还。— —达雷 尔
50、弱者比强者更能得到法律的保护 。—— 威·厄尔

单纯行法,曲线拟合,露天开采,黑白球三位排列游戏

单纯行法,曲线拟合,露天开采,黑白球三位排列游戏

吉林农业大学运筹学课程设计报告年级:2009级专业:信息与计算科学姓名:时间:2011年6月30日目录一、 算法型题目1、单纯形法二、 应用型题目1.曲线拟合2.露天开采3.黑白球三位排列游戏三、 心得体会四、 参考资料一、算法型题目1.实验题目:单纯形法2.设计目的:利用MATLAB 编写单纯型法,熟练掌握MATLAB 编程技巧!3.设计原理:(1) 找出初始可行基,确定初始基可行解,建立初始单纯形表。

(2) 检验各非基变量x j 的检验数是δj =c j -∑=mi 1c bi a ij ,若δj ≤0,j=m+1,…,n则已得到最优解,可停止计算。

否则转入下一步。

(3) 在δj >0,j=m+1,…,n 中,若有某个δk 对应的系数列向量p k ≤0,则此问题是无界,则停止计算。

否则,转入下一步。

(4) 根据max (δj >0)=δk ,X=linprog(C ,A ,B)确定x k 为转入变量,按θ规则计算θ=min (ik i a b ik a >0)=lkl a b 可确定x l 为患处变量,转入下一步。

(5) 以a lk 为主元素进行迭代(即用高斯消去法或称旋转运算),把x k 所对应的列向量P k ⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛mk lk a a k 2k 1a a −−→−变换为⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛0100 −−←第l 行 将X B 列中的x l 换为x k ,得到新的单纯形表。

重复(2)~(5),直到终止。

符号说明:A 表示标准型中的变量的系数矩阵,c 表示目标函数中的基变量的价值系数,b 表示约束方程组右端的常数,t 表示检验数,p 表示表示确定换入变量后计算所得的数值;代码如下:其中A表示标准型中的变量的系数矩阵,c表示目标函数中的基变量的价值系数,b表示约束方程组右端的常数,t表示检验数,p表示表示确定换入变量后计算所得的数值;单纯型法function [x,A,b]=fun(A,c,b,cb)for i=1:size(A,1)x=size(A,2)-size(b)+i-2;endt=[1,1,1,1,1];while ~all(t)<=0%计算检验数for i=1:size(A,2)sum=0;for j=1:size(A,1)sum=sum+A(j,i)*cb(j);endt(i)=c(i)-sum;end%计算最后的bk=find(t==max(t));for h=1:size(A,1)if A(h,k)>0p(h)=b(h)/A(h,k);elseif A(h,k)<=0p(h)=abs(inf);endends=find(p==min(p));%huanru=A(s);%A(s,k);for l=1:size(A,1)q=A(s,k);if l~=sw=A(l,k)/A(s,k);A(l,:)=A(l,:)-A(l,k)/A(s,k)*A(s,:);b(l)=b(l)-w*b(s);else A(s,:)=A(s,:)/q;1b(s)=b(s)/q;endendcb(s)=c(k);x(s)=k;end二、应用型题目1.曲线拟合一,实验题目:曲线拟合二,设计目的:对所给数据,利用ampl对数据进行规范求解,再利用matlab 进行线性拟合,多项式拟合。

wolfe算法确定步长的牛顿法

wolfe算法确定步长的牛顿法
1 3、Wolfe 线性搜索准则为:给定常数 0 1 , 1 2 1 ,取 k 0 使得 2
f ( xk k d k ) f ( xk ) 1 k f ( xk )T d k .....(1) T T f ( xk k d k ) d k 2f ( xk ) d k ......(2)
2
强练习。 三、指导教师评语及成绩: 评语等级 评 语 优 良 中 及 格 不及格
1.实验报告按时完成,字迹清楚,文字叙述流畅,逻辑性强 2.实验方案设计合理 3.实验过程(实验步骤详细,记录完整,数据合理,分析透彻) 4 实验结论正确.

绩: 指导教师签名: 批阅日期:
附录 1:源 程 序
3
function y=fun(x); y=(1/3)*(x(1)^3-x(2)^3-x(2)^2-x(1); function g=gfun(x); g=[x(1)^2-1, x(2)^2-2*x(2)]; function He=Hess(x); n=length(x); He=zeros(n,n); He=[2*x(1),0,0,2*x(2)-2]; function [x,val,k]=dampnm(fun,gfun,Hess,x0); maxk=100; rho=0.5; sigma=0.4; k=0; epsion=le-5; while(k<maxk) gk=feval(gfun,x0); Gk=feval(Hess,x0); dk=Gk\gk; if(norm(dk)<epsion) break; end; m=0; mk=0; while(m<20) if(feval(fun.x0+rho^m*dk)<feval(fun,xo)+sigma*rho^m*gk*dk) mk=m; break; end m=m+1;

线性代数案例

线性代数案例

线性代数案例Cayler-Hamilton 定理【实验目的】1.理解特征多项式的概念2.掌握Cayler-Hamilton 定理 【实验要求】掌握生成Vandermonde 矩阵的vander 命令、求矩阵特征多项式系数的poly()命令、求矩阵范数的norm 命令及矩阵多项式运算的polyvalm 命令 【实验内容】Cayler-Hamilton 定理是矩阵理论中的一个比较重要的定理,其内容为:若矩阵A 的特征多项式为1121)det()(+-++++=-=n n n n n a s a s a s a A sI s f则有()0,f A =亦即11210n n n n a A a A a A a E -+++++=假设矩阵A 为Vandermonde 矩阵,试验证其满足Cayler-Hamilton 定理。

【实验方案】Matlab 提供了求取矩阵特征多项式系数的函数poly(),但是poly()函数会产生一定的误差,而该误差在矩阵多项式求解中可能导致了巨大的误差,从而得出错误的结论。

在实际应用中还有其他简单的数值方法可以精确地求出矩阵的特征多项式系数。

例如,下面给出的Fadeev-Fadeeva 递推算法也可以求出矩阵的特征多项式。

()1111,1,2,...,,,2,...,kk k k k c tr AR k n k R I R AR c I k n--⎧=-=⎪⎨⎪==+=⎩该算法首先给出一个单位矩阵I ,并将之赋给1R ,然后对每个k 的值分别求出特征多项式参数,并更新k R 矩阵,最终得出矩阵的特征多项式的系数k c 。

该算法可以直接由下面的Matlab 语句编写一个()1poly 函数实现:Function c=poly1(A) [nr,nc]=size(A);if nc==nr % 给出若为方阵,则用Fadeev-Fadeeva 算法求特征多项式 I=eye(nc); R=I; c=[1 zeros(1,nc)];for k=1:nc,c(k+1)=-1/k*trace(A*R);r=A*R+c(k+1)*I; endelseif (nr==1 \ nc==1) % 给出为向量时,构造矩阵A=A(isfinite(A));n=length(A) ; % 出去非数或无界的特征根c=[1 zeros(1,n)];for j=1:nc(2:(j+1))=c(2:(j+1))-A(j).*c(1:j);endelse % 参数有误则给出错误信息error (’Argument must be a vector or a square matrix.’)end.【实验过程】>> A = vander([1 2 3 4 5 6 7]);运行结果:A =1 1 1 1 1 1 164 32 16 8 4 2 1729 243 81 27 9 3 14096 1024 256 64 16 4 115625 3125 625 125 25 5 146656 7776 1296 216 36 6 1117649 16807 2401 343 49 7 1 >> A运行结果:aa1 =1.0e+009 *0.0000 -0.0000 -0.0002 0.0287 1.1589 -6.2505 -2.4223 0.0249如调用新的poly1()函数,则可以得出如下的精确结果。

《Legendre小波法求解三类分数阶微积分方程组》

《Legendre小波法求解三类分数阶微积分方程组》

《Legendre小波法求解三类分数阶微积分方程组》篇一一、引言在现代科学研究中,分数阶微分方程作为描述自然现象的重要数学工具,具有广泛的应用场景。

其求解方法也成为了众多学者研究的热点。

本文将介绍一种新的求解方法——Legendre小波法,用于求解三类分数阶微积分方程组。

二、问题背景在现实生活中,存在着大量的复杂问题,需要用到分数阶微分方程进行描述。

这些方程通常涉及到各种物理、化学、生物和工程问题。

然而,由于分数阶微分方程的复杂性,其求解一直是一个挑战。

本文的目的是通过Legendre小波法来有效地求解三类分数阶微积分方程组。

三、Legendre小波法Legendre小波法是一种基于Legendre小波的数值计算方法。

其基本思想是将待求解的函数用Legendre小波进行展开,然后通过一定的方法将分数阶微分方程转化为代数方程组,从而求解出原函数的近似解。

四、求解三类分数阶微积分方程组1. 一维分数阶微分方程组对于一维的分数阶微分方程组,我们首先将Legendre小波作为基函数,对原函数进行展开。

然后,通过利用分数阶微分的性质和Legendre小波的递推关系,将原方程转化为代数方程组。

最后,通过求解代数方程组得到原函数的近似解。

2. 二维分数阶微分方程组对于二维的分数阶微分方程组,我们首先将二维空间进行划分,然后在每个小区域内分别应用一维的Legendre小波展开法。

接着,通过将各个小区域的解进行组合,得到整个区域的解。

同样地,我们利用分数阶微分的性质和Legendre小波的递推关系,将原方程转化为代数方程组,并求解得到近似解。

3. 非线性分数阶微分方程组对于非线性的分数阶微分方程组,我们首先对非线性项进行线性化处理,然后应用Legendre小波法进行求解。

在求解过程中,我们需要注意保持一定的迭代精度和收敛性,以获得稳定的解。

五、实验结果与分析本文通过一系列的实验验证了Legendre小波法在求解三类分数阶微积分方程组中的有效性。

追赶法解三对角方程组

追赶法解三对角方程组

追赶法解三对角方程组(总13页)--本页仅作为文档封面,使用时请直接删除即可----内页可以根据需求调整合适字体及大小--《数值分析》课程设计追赶法解三对角方程组院(系)名称信息工程学院专业班级 10普本信计学号 4学生姓名刘银朋指导教师张荣艳2013 年 05 月 31日数值分析课程设计评阅书课程设计任务书2012—2013学年第二学期专业班级: 10普本信息与计算科学学号: 4 姓名:刘银朋课程设计名称:数值分析Ⅰ、Ⅱ设计题目:追赶法解三对角方程组完成期限:自 2013 年 05月 21 日至2013年05 月31日共 10天设计依据、要求及主要内容:一、设计目的理解追赶法,掌握追赶法的算法设计以及关于追赶法的分析和综合应用,能够较熟练的应用Matlab软件编写求解追赶法的程序和应用Matlab软件数据库软件.二、设计内容(1)认真挑选有代表性的三对角方程组.(2)认真梳理解三对角方程组的解题思路.(3)比较追赶法和高斯消去法的计算精度.三、设计要求1.先用Matlab数据库中的相应的函数对选定的方程,求出具有一定精度的解. 2.然后使用所用的方法编写Matlab程序求解.3.对于使用多个方程解同意问题的,在界面上要设计成菜单的形式.计划答辩时间:2013年 06 月 5 日工作任务鱼工作量要求:查阅文献资料不少于3篇,课程设计报告1篇不少于3000字.指导教师(签字):教研室主任(签字):批准日期: 2013 年 05 月 20 日追赶法解三对角方程组摘要本文主要通过运用追赶法来求解三对角方程组的问题.追赶法是用来求解三对角方程组的专用方法,对于三对角方程组,追赶法比Gauss消去法的计算量要小的多,本文主要介绍了追赶法的原理,并用Matlab编写求解程序,以实现对三对角方程组的求解,进一步解决实际中的问题.并且根据所得出的结果分析追赶法算法和高斯消去的法的计算精度.关键词:追赶法,三对角方程组,追赶法的Matlab程序目录1.前言 ................................................................................................................. 错误!未定义书签。

多边形分解方法计算人体体表局部损伤面积

多边形分解方法计算人体体表局部损伤面积

多边形分解方法计算人体体表局部损伤面积
张必明;熊平
【期刊名称】《中国法医学杂志》
【年(卷),期】2007(022)003
【摘要】在人体体表损伤鉴定时,常常需要测量损伤区域面积、创口长度以及关
节活动角度等指标。

因此,如何准确、方便地测量上述指标一直是法医界关注和研究的课题。

本文采用多边形分解方法,利用数码相机拍摄体表损伤区域图像,使用自主开发的“法医鉴定图像分析测量”软件,测量体表损伤区域面积。

经实验验证,本方法能够达到司法鉴定要求的测量精度。

【总页数】2页(P192-193)
【作者】张必明;熊平
【作者单位】湖南涉外经济学院计算机科学与技术学部,湖南长沙410205;中南大
学信息物理工程学院生物医学工程研究所,湖南长沙410083
【正文语种】中文
【中图分类】D9
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EIGRP分解实验三
1.实验EIGRP 跨路由器建立邻居关系:
a.R2上no EIGRP进程
b.静态路由先让R1 R3之间互相ping通,R1 R3单播互指邻居,注
意出现的提示信息,分析产生的原因
c.将R1 R3接口地址修改为/16,观察收到路由情况,并分析ping
相互收到的路由能否Ping通
2.如图所示,将所有串口和环回口宣告进EIGRP,在R1 R2 R3上开启
debug eigrp packets query reply ack
然后Shut down R1 LO0口,观察各路由器的query包,reply包的发送和确认情况,理解DUAL的扩散查询行为
3.通过在R2上S1/1口设置ACL,阻止R3过来的所有EIGRP包,通
过设置holddown time到最长,使它们之间邻居关系保持不DOWN,然后shut down R1 LO0 口,观察EIGRP的拓扑表,学习EIGRP的SIA状态,以及cisco设计SIA-QUERY/SIA-REPLY两种报文的意义
4.实验解决SIA的方法一,设定ACTIVE-TIME时间;方法二,汇总路

5.实验EIGRP的STUB路由器,理解STUB的几个选项的意义
a.将R3设置成STUB receive-only, R1上观察R3的环回口路由
b.在R3上增加一条静态路由33.3.3.0 指向null0, 通过重分布发
布进EIGRP,观察stub redistributed/static的行为
c.还原,将R2设置成stub,使用ACL/route-map 匹配1.1.1.0/24,
观察stub leek-map选项的行为
6.在R1 R2之间开启EIGRP 的认证
7.在R1上使用distance 255的方法过滤33.3.3.0/24,观察实验现象,
理解没有生效的原因
8.在R3上使用distribute-list 过滤这条33.3.3.0/24,使之在R1上不
可见,但是不能使用基于接口的过滤。

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